Math 272x - Diffeomorphisms of Disks

Lectures by Alexander KupersNotes by Dongryul Kim

Fall 2017

This course was taught by Alexander Kupers. The lectures were on Mondays,Wednesdays, and Fridays at 1–2pm. There was one final paper, and 23 studentswere enrolled. Lectures notes, compiled into a book, can be found on the coursewebsite. (Warning: these notes, taken in class, are incomplete, because I havestopped going to lectures at some point.)

Contents

1 August 30, 2017 41.1 Diffeomorphisms of disks . . . . . . . . . . . . . . . . . . . . . . . 41.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 September 1, 2017 62.1 Cr-manifolds and Cr-maps . . . . . . . . . . . . . . . . . . . . . 62.2 The Whitney topology . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Diffr∂(D1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 September 6, 2017 93.1 Comparing diffeomorphism groups . . . . . . . . . . . . . . . . . 93.2 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Application to gluing . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 The first comparison . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 September 8, 2017 124.1 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Tubular neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . 134.3 The second comparison . . . . . . . . . . . . . . . . . . . . . . . 13

5 September 11, 2017 155.1 Whitney embedding theorem . . . . . . . . . . . . . . . . . . . . 155.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 The third comparison . . . . . . . . . . . . . . . . . . . . . . . . 165.4 Diffeomorphism groups as simplicial groups . . . . . . . . . . . . 17

1 Last Update: January 15, 2018

6 September 13, 2017 186.1 Squares instead of disks . . . . . . . . . . . . . . . . . . . . . . . 186.2 Smale’s theorem on Diff∂(D2) . . . . . . . . . . . . . . . . . . . . 186.3 Diffeomorphisms of S2 . . . . . . . . . . . . . . . . . . . . . . . . 20

7 September 15, 2017 217.1 Isotopy extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Parametrized isotopy extension . . . . . . . . . . . . . . . . . . . 22

8 September 18, 2017 248.1 Embeddings of Rm . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Connected sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Diffeomorphisms of S2 . . . . . . . . . . . . . . . . . . . . . . . . 26

9 September 20, 2017 279.1 Gramain’s proof of Diff∂(D2) ' ∗ . . . . . . . . . . . . . . . . . . 279.2 Space of arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3 Path components of Diff∂(Σ) . . . . . . . . . . . . . . . . . . . . 29

10 September 22, 2017 3010.1 Restating Smale’s theorem . . . . . . . . . . . . . . . . . . . . . . 3010.2 Hatcher’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 September 25, 2017 3211.1 Sard’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

12 September 27, 2017 3512.1 Jet transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13 September 29, 2017 3813.1 Morse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.2 Morse lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 October 2, 2017 4114.1 Generic Morse function . . . . . . . . . . . . . . . . . . . . . . . 4114.2 Understanding level sets . . . . . . . . . . . . . . . . . . . . . . . 4114.3 Handle decomposition . . . . . . . . . . . . . . . . . . . . . . . . 43

15 October 4, 2017 4415.1 Cobordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.2 Handle isotopy, rearrangement, and cancellation . . . . . . . . . 45

16 October 6, 2017 4716.1 Handle decompositions and the topology . . . . . . . . . . . . . . 4716.2 Handle exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816.3 Removing 0, 1, w − 1, w-handles . . . . . . . . . . . . . . . . . . 48

2

17 October 11, 2017 5017.1 Two-index lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.2 Wh1(π1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5117.3 Poincare conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 52

18 October 13, 2017 5318.1 A Brieskorn sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.2 Signature of the sphere . . . . . . . . . . . . . . . . . . . . . . . . 55

19 October 16, 2017 5719.1 Whitney trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.2 Casson trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

20 October 18, 2017 6020.1 Motivation for algebraic K-theory . . . . . . . . . . . . . . . . . 6020.2 Group completion . . . . . . . . . . . . . . . . . . . . . . . . . . 60

21 October 20, 2017 6321.1 S•-construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6321.2 Algebraic K-theory of spaces . . . . . . . . . . . . . . . . . . . . 6421.3 Moduli spaces of h-cobordisms . . . . . . . . . . . . . . . . . . . 65

22 October 23, 2017 6722.1 Computation of π0H(X) . . . . . . . . . . . . . . . . . . . . . . . 6722.2 Computation of π1H(X)—the HWI sequence . . . . . . . . . . . 68

23 October 25, 2017 7023.1 Pseudo-isotopy implies isotopy . . . . . . . . . . . . . . . . . . . 7023.2 π0(Diff∂(Dn)) and Θn+1 . . . . . . . . . . . . . . . . . . . . . . . 70

24 October 27, 2017 7224.1 Concordances rationally . . . . . . . . . . . . . . . . . . . . . . . 7224.2 Block diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 7224.3 Hatcher spectral sequence . . . . . . . . . . . . . . . . . . . . . . 7324.4 Farrel–Hsiang theorem . . . . . . . . . . . . . . . . . . . . . . . . 74

25 October 30, 2017 7525.1 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7525.2 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7625.3 Kirby–Siebenmann bundle theorem . . . . . . . . . . . . . . . . . 76

26 November 1, 2017 7926.1 Verifying the fiberwise engulfing condition . . . . . . . . . . . . . 7926.2 Proof of Kirby–Siebenmann bundle theorem . . . . . . . . . . . . 8026.3 Space of smooth structures . . . . . . . . . . . . . . . . . . . . . 81

3

Math 272x Notes 4

1 August 30, 2017

1.1 Diffeomorphisms of disks

The main object of interest is the topological group of diffeomorphisms of adisk. We have

Dn = x ∈ Rn : ‖x‖ ≤ 1, ∂Dn = Sn−1 = x ∈ Rn : ‖x‖ = 1.

We then define Diff∂(Dn) as the topological group of (smooth) diffeomorphismsof Dn that fix ∂Dn pointwise, in the C∞-topology.

Question. What is the homotopy type of Diff∂(Dn)?

We certainly want to compute the path components, isotopy classes. Alsowe might want to compute homotopy or homology groups. It turns out thatthis is related to algebraic K-theory.

There are several reasons for studying this object. Firstly, manifolds areinteresting and so diffeomorphisms are also interesting. All manifolds are builtout of disks. Also, disks capture the difference between “local” and “global”phenomena. In Rn you might be able to use some Eilenberg swindle to pushproblems away to infinity. Considering only diffeomorphisms capture the dif-ference between “smooth” and “continuous”. For instance, it is trivial to studyhomeomorphisms of the disk.

These objects also have interactions with other fields. As I have mentioned,surgery theory allows us to relate Diff∂(Dn) to homotopy theory and algebraicK-theory. They also have connections with cobordism categories and graphcomplexes.

1.2 Overview

I plan to start with low dimensions, dimensions at most 3. Define

M∂(Dn) =∐[σ]

BDiff∂(Dnσ),

where [σ] ranges over isotopy classes of smooth structures on Dn, standard nearthe boundary.

Theorem 1.1 (Folhlove, Rado, Perelman). π0M∂(Dn) ' ∗.

Theorem 1.2 (Folhlove, Smale, Hatcher). Diff∂ Dn ' ∗.

Next we go to high dimensions, when n ≥ 5. The main tool here is thes-cobordism theorem.

Definition 1.3. If M0 and M1 are closed n-dimensional manifolds, then acobordism between them is a compact (n + 1)-dimensional manifold N with∂N ∼= M0 qM1. If M0 → N and M1 → N are weak equivalences, then N issaid to be an h-cobordism.

Math 272x Notes 5

Theorem 1.4. An h-cobordism N is diffeomorphism to M0 × I if and only ifτ(N) ∈Wh1(π1M0) vanishes.

Here this Whitehead group is the quotient of K1(Z[π1(M0)]). These lead tothe computations

π0M∂(Dn) ∼= Θn, π0 Diff∂(Dn) ∼= Θn+1.

These are groups of homotopy spheres and a famous paper of Kervaire andMilnor relate these to stable homotopy groups of spheres.

There is an additional structure on Diff∂(Dn). If f1, . . . , fk ∈ Diff∂(Dn) andthere is an embedding e : qkDn → Dn, then you can use this to get a structureof an En-algebra. It follows from this that BDiff∂(Dn) ' ΩnX for some spaceX.

Theorem 1.5 (Molvet). BDiff∂(Dn) ' Ωn0 Top(n)/O(n), where Top(n) is thehomeomorphisms of Rn fixing the origin.

After this we will talk about some quite recent work. Consider a categoryCob(n) that has objects the moduli spaces of (n + 1)-dimensional manifolds,and morphisms the moduli spaces of n-dimensional cobordism.

Theorem 1.6 (Galatius–Madsen–Tillmann–Weiss). B Cob(n) ' Ω∞−1MTO(n).

Theorem 1.7 (Galatius–RAndal–Williams). For the space Wg,1 = (#gSn ×

Sn) \ int(D2n), The map BDiff∂(Wg,1) → Ω∞−1MTΘ is an H∗-isomorphismfor ∗ ≤ (g − 3)/2.

There is a sequence

Diff∂(Wg,1)→ Emb'12∂

(Wg,1)→M∂(D2n)

and using the information on the first two, we can get information about thelast one. Here, Emb'1

2∂is the embeddings that fix half the boundary, and we

can use embedding calculus to get information.

Math 272x Notes 6

2 September 1, 2017

We are going to do a review of differential topology. We are also going to showthat Diff∂(D1) is contractible.

2.1 Cr-manifolds and Cr-maps

Definition 2.1. A d-dimensional topological manifold is a second-countableHausdorff topological space X that is locally homeomorphic to on open subsetof Rd.

Then you can make an “atlas” consisting of “charts” X ⊇ Viφi−→ Wi ⊆ Rd

with transition functions φjφ−1i : φi(Vi ∩ Vj)→Wj .

Definition 2.2. For r ∈ N ∪ ∞, a d-dimensional Cr-manifold is a second-countable Hausdorff topological space X with the data of maximal atlas ofcharts with Cr transition functions.

For manifolds with boundary, you replace Rd with Rd−1 × [0,∞). Thenthere is a subset ∂X ⊆ X consisting of points p ∈ X going to Rd−1 ×0. Thisis well-defined, i.e., a point can’t go to both the boundary and the interior ofRd−1×0. You can check this by computing the homology H∗(U,U \ p). Thiswill also show that ∂X will be a Cr-manifold of dimension d− 1.

There are possibly two notions of Cr-functions on Rd−1 × [0,∞).

• extend to Cr-functions on an open neighborhood

• Cr on the interior and it and its partial derivatives extend continuouslyto Rd−1 × 0

Luckily a theorem of Whitney states that they are equivalent.

2.2 The Whitney topology

This is a topology on Cr(M,N) and there are definitions in terms of a sub-basisand as a subspace of sections. There are weak Whitney and strong Whitneytopologies, but they coincide for compact manifolds.

By “convergence”, we mean uniform convergence of functions and the firstr derivatives on compact subsets.

Definition 2.3. For r ∈ N, the (weak) Whitney topology on Cr(M,N) isthe topology generated by the sub-basis consisting of

Nr(f, φ, ψ,K, ε) =

where f : M → N is a Cr-map, φ : V → W ⊆ Rm and ψ : V ′ → W ′ ⊆ Rn arecharts in M and N , K ⊆ V is compact such that f(K) ⊆ V ′, and ε > 0. Thisset consists of g ∈ Cr(M,N) such that g(K) ⊆ V ′ and

|DI(ψfφ−1)k(x)−DI(ψgφ−1)k(x)| < ε

Math 272x Notes 7

for all x ∈ K, |I| ≤ r, and 1 ≤ k ≤ n.For r = ∞, the topology on C∞(M,N) is the coarsest one such that

C∞(M,N) → Cr(M,N) is continuous for all r <∞.

We can also define as a subspace of a section space. Let s ≤ r. Then theset of s-jets of Cr functions f : Rm → Rn is the quotient of Cr(Rm,Rn) by theequivalence relation

f ∼s g ⇔ DIfk(0) = DIgk(0) for all |I| ≤ s and 1 ≤ k ≤ n.

We can identify Js(Rm,Rn) = Cr(Rm,Rn)/ ∼s with n-tuples of real polynomi-als of degree at most s in m variables. This is Taylor expansion. So this is afinite-dimensional real vector space and there is a standard topology on here.

This generalizes to Cr(M,N). Pick a point p ∈M and define an equivalencerelation f ∼s,p g if f(p) = g(p) and the first s partial derivatives coincide oncharts.

Definition 2.4. The set of s-jets of Cr-maps M → N is defined as

Jr(M,N) = (M × Cr(M,N))/ ∼

where (p, f) ∼ (q, g) if p = q and f ∼s,p g.

Then there is a natural projection π : Js(M,N) → M and an evaluationτ : Js(M,N)→ N . So we can combine them to get

(π, τ) : Js(M,N)→M ×N.

Using charts, we see that this map locally looks like Rm ×Rn × Js0 (Rm,Rn)→Rm×Rn. We can use this to topology Js(M,N) so that π : Js(M,N)→M×Nis locally trivial.

For a Cr-map f : M → N we can define

js(f) : M(id,f)−−−→M × Cr(M,N)

quotient−−−−−→ Js(M,N).

This is a section, and so we have an embedding

js : Cr(M,N) → Γ(M,Js(M,N))

into the space of sections. Now topologize Cr(M,N) by taking the subspacetopology on

Cr(M,N) → Γ(M,Jr(M,N)) → Map(M,Jr(M,N))

with the compact-open topology.

Proposition 2.5. This topology is equal to the Whitney topology.

The advantage of this is that you can use various adjunctions of the compact-open topologies to show that certain maps are continuous.

Corollary 2.6. Cr(M,N) → Cr−1(M,N) is continuous, because the projectionJr(M,N)→ Jr−1(M,N) is continuous. Likewise the composition Cr(M,N)×Cr(N,P )→ Cr(M,P ) is continuous. The inverse map on Diffr∂(M) is contin-uous.

Corollary 2.7. Diffr∂(M) are topological groups in the Whitney topology.

Math 272x Notes 8

2.3 Diffr∂(D1)

Theorem 2.8. Diffr∂(D1) ' ∗.

Proof. This is basically a convexity argument. We are explicitly going to defor-mation retract to id, which is given by

H : Diffr∂(D1)× [0, 1]→ Diffr∂(D1); (f, t) 7→ (1− t)f + t idD1 .

For continuity, check that the first r derivatives converge uniformly. Also, thisactually lands in Diffr∂(D1).

Math 272x Notes 9

3 September 6, 2017

Last time I defined the Whitney topology and showed that the diffeomorphismof D1 is trivial. The argument also works for different variants, but fails forhigher dimension.

3.1 Comparing diffeomorphism groups

Let M be a C∞-manifold with boundary ∂M .

Definition 3.1. The group Diffr∂(M) is the Cr-diffeomorphisms of M that fix∂M pointwise. In this group, Diffr∂,D(M) is the Cr-diffeomorphisms of M suchthat f and its first r derivatives coincide with id at ∂M . In this group, thereis Diffr∂,U (M) is the Cr-diffeomorphisms of M that agree with id on an openneighborhood of ∂M .

Example 3.2. I can sheer things in Diffr∂(M). I can only make things point inthe same direction in Diffr∂,U .

Theorem 3.3. All the inclusions in

Diff1∂,U (M) Diff1

∂,D(M) Diff1∂(M)

......

...

Diff∞∂,U (M) Diff∞∂,D(M) Diff∞∂ (M)

are weak equivalences.

3.2 Collars

Definition 3.4. A collar of ∂M in M is an Cr-embedding c : ∂M×[0, 1) →Mthat is identity on ∂M .

We are going to show that they exist. We are going to flow along a vectorfield, which we should show exists.

Definition 3.5. A partition of unity subordinate to an open cover Uii∈Iof a topological space X is a collection of continuous functions ηi : X → [0, 1]such that

(1) supp(ηi) ⊆ Ui for all i,

(2) only finitely many ηi are nonzero at a given p ∈ X,

(3)∑i∈I ηi = 1.

Math 272x Notes 10

Partitions on unity exist on M since M is paracompact, and we can assumethat ηi is Cr.

Definition 3.6. A vector field X on M is said to be inwards pointing if for

all p ∈ ∂M , there is a chart M ⊇ Vϕ−→ W ⊆ Rm−1 × [0,∞) such that in these

coordinates X(p)m > 0.

Note that this is a convex condition; if X and Y are inwards pointing at p,then tX + (1− t)Y is inwards pointing at p for all t ∈ [0, 1].

Lemma 3.7. There exists a Cr-inwards pointing vector field on M .

Proof. Locally we may take ∂/∂xm, and we can pull these back to get vectorfields on Vi inwards pointing. Then we can patch these together by taking a Cr

partition of unity with respect to Vi and write X =∑ηiXi.

Now consider the ODE for γ : R→M , given by

d

dtγ(t) = X(γ(t)).

By Picard–Lindelof, the following is well-defined.

Lemma 3.8. There is an open neighborhood V of ∂M × 0 in ∂M × [0,∞)such that there is a Cr map

ΦX : V →M ; (q, t) 7→ γq(t).

Lemma 3.9. The map ΦX has bijective differential at (q, 0) ∈ ∂M × 0 ⊆ V ,for all q ∈ ∂M .

Proof. This follows from the inwards pointing. The derivative on Tq∂M is goingto be the identity, and the last diagonal entry is X(a)m.

By the inverse function theorem, Φx is a local diffeomorphism.

Lemma 3.10. Suppose that Y is a metric space and f : Y → Z is locally atopological embedding. If X ⊆ Y is a set such that f |X is injective, then thereexists a neighborhood U ⊇ X such that f |U is injective.

Using this point-set lemma, we can shrink the domain V of ΦX to get aCr-embedding.

Theorem 3.11. Collars exist.

If you were writing a differential topology book, what you should have doneis to prove a relative version. These allows you to prove uniqueness results,by extending a collar on ∂M × ∂I to ∂M × I and getting an isotopy. Thenyou would be able to prove that the space of collars, appropriately defined, iscontractible.

Math 272x Notes 11

3.3 Application to gluing

Suppose we are given Cr-manifolds M,N , and a Cr-diffeomorphism ϕ : ∂M →∂N . Using this you can form M ∪ϕ N to get a manifold.

But you don’t really know if this is a Cr-manifold, because you can havecringes at that boundary. Here existence of uniqueness of collar neighborhoodsshow that M ∪ϕ N has a well-defined Cr-manifolds structure.

What you do is both pick collars for them, look at (−1, 0]×∂M and [0, 1)×∂N , and then glue them to (−1, 1)× ∂M , which we know well.

3.4 The first comparison

Theorem 3.12. The embedding Diffr∂,U (M) → Diffr∂,D(M) is a homotopyequivalence.

Proof. There is a collar c : ∂M × [0, 1) → M and then we can form a biggermanifold

M = (−1, 0]× ∂M ∪id M.

This has a bigger collar c : (−1, 1)× ∂M → M .Then Diffr∂,U (M) is homeomorphic to the subgroup of Diffr(M) that are the

identity on a neighborhood of (−1, 0]× ∂M , and Diffr∂,D(M) is homeomorphic

to a subgroup of Diffr(M) that are the identity on (−1, 0]× ∂M .Now we can write out the homotopy inverse by a standard trick of sliding.

Let η : (−1, 1) → (−1, 1) be a Cr-embedding that is the identity near 1, andsending [0, 1) to [ 1

2 , 1). This induces a Cr-embedding M → M

sη(p) =

c(q, η(t)) if p ∈ c(q, t)p otherwise,

and we define the homotopy inverse as

r(f)(p) =

sη f s−1

η (p) if p ∈ im(sη),

p otherwise.

To show that r and i are inverse up to homotopy, use the conjugation withs(1−t)η+t·id.

Math 272x Notes 12

4 September 8, 2017

We showed in the last lecture that Diffr∂,U (M) → Diffr∂,D(M) are homotopyequivalences. Today we are going to show that Diffr∂,D(M) → Diffr∂(M) areweak equivalences.

4.1 The exponential map

This is something that takes a tangent vector and gives the geodesic.

Lemma 4.1. M admits a C∞ Riemannian metric.

Proof. These locally exists, and we combine them using a C∞ partition of unity.

Fix a C∞ Riemannian metric g on M , and for a moment take ∂M = ∅. Thengeodesics will be extrema for the energy functional, defined for γ : [a, b] → Mas

E(γ) = (b− a)

∫ b

a

‖γ′(t)‖2dt.

Definition 4.2. γ is a geodesic if for all a′ < b′ in [a, b], γ|[a′,b′] is a localextremum of E among all C1 paths [a′, b′]→M with the same endpoints.

Variational calculus tells us that γ is a geodesic if and only if it satisfies theEuler–Lagrange equation given by

d2γidt2

= −∑j,k

Γijk(γ(t))dγidt

dγkdt

.

These Γijk are Christoffel symbols and are C∞ functions. Picard–Lindelof im-plies

Lemma 4.3. For all p ∈ M there exists a neighborhood U ⊆ M of p and anε > 0 such that for all q ∈ U and v ∈ TqM with ‖v‖ < ε, there is a uniquegeodesic γ : (−2, 2)→M with γ(0) = q and γ′(0) = v.

Lemma 4.4. There exists an open neighborhood V of the 0-section in TM anda smooth map

Γ : U × (−2, 2)→M

such that Γ|(q,v)×(−2,2) is the unique geodesic.

Definition 4.5. The exponential map exp : U → M is the restriction of Γto U × 1.

This is a C∞ map, and if we compute the derivative at (q, 0) in the 0-section,we get

T(q,0)U ∼= TqM ⊕ TqM → TqM

which is just addition.If ∂M 6= ∅, the above works if we pick a C∞ collar c : ∂M × [0, 1) →M and

take the Riemannian metric of the form

g|∂M = g∂ + dt2.

Math 272x Notes 13

4.2 Tubular neighborhoods

Let M ⊆ N be a Cr submanifold with ∂M = ∅. This has a normal bundleνM = TN |M/TM , which can be naturally identified with (TM)⊥ ⊆ TN |M ifyou have a Riemannian metric.

Definition 4.6. A tubular neighborhood for M in N is a Cr-embeddingΦM : νM → N such that

TM ⊕ νM TN |M

νM νM

DΦM

id

commute.

Theorem 4.7. Every Cr-submanifold M ⊆ N with ∂M = ∅ has a Cr-tubularneighborhood.

Proof. We can identify νM with TM⊥ ⊆ TN |M and apply the exponential mapto the disk bundle DεςM ⊆ νM , where ε is a function not a constant.

Now exp |DενM is going to have the right derivative at the 0-section. By theinverse function theorem this a local Cr-diffeomorphism and so by decreasingε we may arrange it to a Cr-embedding. By fiberwise rescaling, we get νM →DενM → N .

In the case ∂M 6= ∅, this same argument works if M ⊆ N has “neat bound-ary”. This in particular means that ∂M = M ∩ ∂N and the manifold M meatsthe boundary ∂N orthogonally.

4.3 The second comparison

Our goal is to show that Diffr∂,D(M) → Diffr∂(M) is a homotopy equivalence.First pick a collar c : ∂M × [0, 1) → M , and then identify T(q,0)M ∼=

Tq∂M ⊕ ε, where ε is the 1-dimensional space generated by ∂/∂t. To see thedifference between the two groups, let’s write down the derivatives in thesecoordinates.

A diffeomorphism f ∈ Diffr∂,D is going to have first derivative[idTp∂M 0

0 idε

]and g ∈ Diffr∂(M) is going to have first derivative[

idTp∂M χ(q)0 λ(q) idε

]where χ(q) is a Cr-vector field corresponding to sheering and λ(q) is a functioncorresponding to scaling.

Math 272x Notes 14

So we need to remove λ and χ. Pick a continuous

η : (0,∞)→ Diffr∂([0, 1))

such that η(l) is id on [ 12 , 1), (η(`))′(0) = `−1, and η(1) = id. For a Cr-map

λ : ∂M → (0,∞), we define a map

η(λ)(p) =

(q, η(λ)v) p = (q, v) ∈ ∂M × [0, 1)

p otherwise,

and considerη(λ(g)) g ∈ Diffr∂(M).

This is going fix λ to 1 in the derivative at ∂M .Now pick a Cr function ρ : [0, 1)→ [0, 1] that is 1 near 0, 0 on [ 1

2 , 1). GivenY a Cr-vector field on ∂M , we can define

G(y)(p) =

(Γ∂M (q,−Y (q), ρ(t)), t) if p = (q, t) ∈ ∂M × [0, 1)

p otherwise.

This then fixes the χ(q) part in the first derivative on ∂M .By suitably interpolating the η and ρ, we see that the maps are indeed

homotopy inverses.

Math 272x Notes 15

5 September 11, 2017

5.1 Whitney embedding theorem

Theorem 5.1 (Weak Whitney embedding). Every compact Cr-manifold M of∂M admits a Cr-embedding into RN .

Proof. Take Viki=1 an open cover by domains of charts M ⊇ Viϕi−→Wi ⊆ Rm

and ηiki=1 a partition of unity subordinate to this open cover. Define

ϕi : M → Rm+1; x 7→

(ηi(x), ηi(x)ϕi(x)) if x ∈ Vi0 otherwise.

Then use x 7→ (ϕ1(x), . . . , ϕk(x)).

If you be careful, you can show that for large enough N , the space of em-beddings M → RN is contractible. Using the Whitney trick, you can show thatM embeds into R2m.

There is a version for manifolds with boundary. First embed ∂M into RN ,then extend it to ∂M × [0, 1) → RN × [0,∞) and extend by a relative version.

5.2 Convolution

Definition 5.2. For f : Rn → R continuous and compactly supported andg : Rn → Rm continuous, define the convolution

(f ∗ g)(x) =

∫y∈R

f(x− y)g(y)dy.

Differentiation under the integral implies that f ∗ g is smooth if f is. So letη : Rn → [0,∞) that is supported in Dn and

∫y∈Rn η(y)dy = 1. If we define

ηε(x) = 1εn η(xε ), then

ηε ∗ g → g

as ε→ 0 in the Cr-topology if g was Cr.More generally, let M,N be compact smooth manifolds with empty bound-

ary. Take C∞-embeddings ϕM : M → RkM and ϕN : N → RkN and C∞-tubular neighborhoods

ΦM : νM → RkM , ΦN : νN → RkN .

There are then going to be C∞ projection maps

πM : im(ΦM )→M, πN : im(ΦN )→ N.

For f : M → N a Cr function, we can look at

πN

(∫y∈RkM

ηε(y)f(πM (x− y))dy

).

Math 272x Notes 16

This may not make sense, but this will make sense if ε is small enough. If ε issmall, then x − y is in the tubular neighborhood if ηε(y) 6= 0 and other thingsalso make sense. Then

ηε ∗π f : M → N

is a well-defined map and this converges to f in the Cr-topology as ε→ 0.

5.3 The third comparison

We now want to show

Theorem 5.3. Diff∞∂,D(M) → Diffr∂,D(M) are weak equivalences.

Proof. Pick a collar c : ∂M × [0, 1) → M and a neat embedding ϕM : M →RkM−1 × [0,∞) and a tubular neighborhood ΦM : νM → RkM−1 × [0,∞). Thisagain gives a projection map πM : im(ΦM )→M .

Pick a ρ : [0, 1)→ [0, 1] that is 0 on [0, 14 ] and 1 near 1. Given a commutative

diagram

Si Diff∞∂,D(M)

Di+1 Diffr∂,D(M),G

we need to show that we can homotope this so that there is a lift.By sliding the collar, we can assume that all Gs for s ∈ Di+1 are the identity

on ∂M × [0, 2]. Now from ρ : [0, 1)→ [0, 1], we can construct

ρ : M → [0, 1]; p 7→

ρ(t) if p = (q, t) ∈ ∂M × [0, 1)

1 otherwise.

So I’m tuning it down near the boundary. For s ∈ Di+1 define

ηε∗πGs(x) = πM

((1− ρ(x))Gs(x) + ρ(x)

∫y∈RkM

ηε(y)Gs(πM (x− y))dy

)This, if defined, is then going to be a C∞ function, because if ρ(t) 6= 1 thenHs = id. For each s ∈ Di+1 there exists a ε(s) such that this is well-defined anda diffeomorphism. Since Diffr∂,D(M) is open in Cr(M), there is a single ε0 > 0

that works for all s ∈ Di+1.Now our homotopy will be

[0, 1] 3 τ 7→(x 7→ πM

((1−τρ(x))Gs(x)+τρ(x)

∫y∈RkM

ηε(y)Gs(πM (x−y))dy

))This shows that the inclusion is a weak equivalence.

Math 272x Notes 17

5.4 Diffeomorphism groups as simplicial groups

There is of course the singular simplices Sing(Diffr∂(M)) with

Sing(Diffr∂(M))k = ∆k → Diffr∂(M).

But there is also a more geometric way. Define

SDiffr∂(M)k =

∆k ×M ∆k ×M

∆k

fixing ∆k × ∂M and Cr

which is strictly smaller than Sing(Diffr∂(M)), because we have the additionalCr condition. But you can show that the inclusion is a weak equivalence.

Math 272x Notes 18

6 September 13, 2017

We have seen that the differentiability conditions really don’t matter. So fromnow on, all diffeomorphisms and embeddings are C∞. Today we are going toshow that Diff∂(D2) ' ∗ by using Smale’s “dynamical proof”. Other proofsinclude

• Cerf/Gramain using “algebraic topology”,

• Hatcher using “parametrized Morse theory”,

• complex-analytic proof,

• curve-shortening flow.

6.1 Squares instead of disks

For Smale’s proof, it’s better to use squares instead of disks. To compare them,we use the following construction, If M and M ′ are n-dimensional manifoldswithout boundary, and we have an embedding i : M → M ′, then there is acontinuous map

i∗ : Diff∂,U (M) → Diff∂,U (M ′); f 7→(p 7→

p p /∈ i(M)

i(f(i−1(p))) p ∈ i(M)

).

If i is isotopic to i′, then (i)∗ to (i′)∗ are homotopic.This means that if we also have an embedding j : M ′ → M such that i j

is isotopic to idM and j i is isotopic to idM , then (i)∗ and (j)∗ are homotopyinverses.

Proposition 6.1. Diff∂,U (I2) and Diff∂,U (D2) are homotopy equivalent.

Proof. Take the scaling embedding that puts the circle in the square and squarein the circle.

6.2 Smale’s theorem on Diff∂(D2)

So it suffices to prove Diff∂,U (I2) ' ∗. The idea is to take, for each f ∈Diff∂,U (I2), the vector field on I2 given by pushing forward ~e1 along f :

χ(f)(x, y) = Dff−1(x,y)(~e1).

We can reconstruct f from this vector field χ(f). To see this, for convenienceextend χ(f) to a vector field on R2 by ~e1. Then the flow Φχ(f) : R× R2 → R2

is well-defined. If we take 0× y ∈ 0× I and flow from time t along χ(f),we get

ϕ(t) = Φχ(f)(t, 0, y) = f(t, y)

by looking at the differential equations they solve.

Math 272x Notes 19

This χ(f) is not an arbitrary vector field, but has to be in

S = smooth everywhere nonzero vector fields on I2 that is ~e1 on ∂I2.

On the other hand, every vector field Y ∈ S can be used to construct a diffeo-morphism of I2.

Lemma 6.2. Let Y ∈ S. For (0, y) ∈ 0 × I, there exists a unique τ(Y, y) ∈(0,∞) such that ΦY (τ(Y, y), 0, y) ∈ 1 × I. This time τ(Y, y) depends contin-uously on Y and smoothly of y.

Proof. Uniqueness is clear, by just extending the vector field. Continuity of Yand smoothness of y follows from dependence of solutions of differential equa-tions to initial conditions and coefficients.

The hard part is existence. The flowline γ, if it leaves I2, has to leavethrough 1 × I. So if τ(Y, y) does not exist, it has to stay inside the square.Consider the forwards limiting set

I2 ⊇ σ∗(γ) =⋂α>0

γ([α,∞)).

Poincare–Bendixson theorem says that ∞∗ is one of the following:

(1) a fixed point of the flow

(2) a path between fixed points

(3) periodic orbit

Because the vector field is non-vanishing, (1) and (2) are ruled out. The periodicorbit is also ruled out by similar reasons. If there is a periodic orbit, then bysomething like Brouwer fixed-point theorem there has to be a fixed point.

Now you would be tempted to write down

ΨY : I2 → I2; ΨY (x, y) = ΦY (τ(Y, y)x, 0, y),

but this is not the identity near the boundary.So we tune this down by bump functions. Pick a smooth family of smooth

embeddings ητ : [0, 1]→ [0,∞) defined for τ ∈ (0,∞) such that

(i) ητ (0) = 0, ητ ([0, 1]) = [0, τ ],

(ii) ητ has derivative 1 near 0 and 1,

(iii) η1 = id.

Now we write

ΨY : I2 → I2; ΨY (x, y) = ΦY (ητ(Y,y)(y), 0, y).

Proposition 6.3. ΨY is a diffeomorphism of I2 which is id near ∂I2.

Math 272x Notes 20

Proof. It is a composition of (x, y) 7→ (ητ(Y,y)(x), 0, y) and (t, y) 7→ ΦY (t, 0, y).

Furthermore, we get a continuous map Ψ : S → Diff∂,U (I2), and for χ(f),we get Ψχ(f) = f . So to prove Smale’s theorem, it suffices to prove that Sdeformation retracts onto χ(id) = ~e1.

Let’s think of S as smooth maps I2 → R2 \ 0 which equals ~e1 near theboundary. Let U denote the universal cover of R2 \ 0. Then we can uniquely

lift Y ∈ S to Y : I2 → U by specifying that Y (0, 0) = ~f1. Note that U is

contractible, and hence we can find a deformation retraction onto ~f1. Thisinduces a deformation retraction of S onto ~e1.

Theorem 6.4 (Smale). Diff∂(D2) ' ∗

6.3 Diffeomorphisms of S2

There is certainly a map O(3) → Diff(S2), and you have

Diff(S2)→ FrGL(TS2)→ FrO(TS2) ∼= O(3).

You can check that it follows Diff(S2) = O(3) × G, and this G will be con-tractible.

Math 272x Notes 21

7 September 15, 2017

Today I am going to talk about the parametrized isotopy extension theorem.This is the beginning of actually proving things like Diff(S2) ' O(3) or theGromain/Cerf’s proof of Smale’s theorem.

7.1 Isotopy extension

Before doing a parametrized case, let’s do a non-parametrized case. Given apath γ : [0, 1] → M , is there a path of diffeomorphisms inducing it? This isintuitively clear; you assume that the space is made out of flexible material, putyour finger at the point and drag along the curve.

Definition 7.1. An isotopy of embeddings M → N is a neat embedding

g : M × [0, 1] → N × [0, 1]

that preserves the [0, 1] coordinate. An isotopy of diffeomorphisms is adiffeomorphism

f : N × [0, 1]→ N × [0, 1]

that preserves the [0, 1] coordinate.

In these cases, gt = g|M×t : M → N is an embedding and ft = f |N×t :N ×N is a diffeomorphism.

Alternative definitions include g : M × [0, 1]→ N being smooth and gt is anembedding for all t. But this is slightly different. You could also say that anisotopy is a map g : [0, 1]→ Emb(M,N), but the problem is that M×[0, 1]→ Nmight not be smooth. So we are not going to use these definitions.

Question. Given an isotopy of embeddings g : M × [0, 1] → N × [0, 1], whendoes there exist an isotopy of diffeomorphisms f : N × [0, 1] → N × [0, 1] suchthat gt = ft g0?

The answer is yes if g is compactly supported away from ∂M , i.e., thereexists a compact K ⊆M \ ∂M such that gt|M\K = g0|M\K .

Theorem 7.2. For g : M × [0, 1] → N × [0, 1] a compactly-supported isotopyof embeddings, there exists a compactly-supported isotopy of diffeomorphisms

f : N × [0, 1]→ N × [0, 1]

such that f0 = id and gt = ft g0.

Proof. Look at the vector field ∂/∂t on M× [0, 1]. We can push it forward alongg to get a smooth section χ(g) of T (N × [0, 1])|g(M×[0,1]). Then the flow alongχ(g) for time t starting at g(p, 0) will end up at g(p, t) = (gt(p), t).

If we can extend χ(g) to a vector field χ(f) on N×[0, 1] such that π∗(χ(f)) =∂/∂t, then we can attempt to flow along χ(f) to define f . This will be compactly-supported and exist if χ(f) equals ∂/∂t (on N × [0, 1]) outside of a compact

Math 272x Notes 22

set and π∗(χ(f)) = ∂/∂t. To construct this, it actually suffices that π∗(χ(f)) isa positive multiple of ∂/∂t on a compact subset. After this we can fix this byrescaling.

We will produce χ(f) locally and patch these together by a partition of unity.Take the finitely many charts covering the image of the support of g. Let thesebe

ϕ : N × [0, 1] ⊇ Vi →Wi ⊆ Rn−1 × [0,∞)

such that ϕi(g(M × [0, 1])∩ Vi) = (Rm−1 × [0,∞))∩Wi. To extend χ(g)|Vi , wecan instead extend

(ϕi)∗(g∗)( ∂∂t

)can we can do this by making it constant in the remaining (n−m) directions.Pull them back to Vi using ϕi to get χi(f) on Vi. Now π∗(χi(f)) is equal to∂/∂t on g(M × [0, 1]) ∩Ci. By shrinking Vi if necessary, we can guarantee thatπ∗(χi(f)) is a positive multiple of ∂/∂t.

Now take V0 ⊆ N × [0, 1] open outside g(M × [0, 1], so that V0, V1, . . . , Vk isan open cover of N × [0, 1]. Now take a partition of unity ηiki=0 with respectto this cover, and

χ(f) = η0 ·∂

∂t+

k∑i=1

ηiχi(f)

works.

Example 7.3. Without the compact-support condition, this fails. For instance,take a knot in R3 with ends at infinities. Consider an isotopy that in time [0, 1]moves it to ∞. There can’t be an isotopy of diffeomorphisms extending them,because the embeddings at t = 0 and t = 1 are not even diffeomorphic.

7.2 Parametrized isotopy extension

The proof generalizes to the following. You could either do this inductively orprove similarly.

Theorem 7.4. Given a k-parameter family of embeddings g : M×∆k → N×∆k

which has compact support K ⊆ M \ ∂M , there exists a compactly supportedk-parameter isotopy of diffeomorphisms f : N ×∆k → N ×∆k such that gt =ft g(1,0,...,0).

Definition 7.5. The simplicial set SEmb(M,N) is defined as k-simplices givenby k-parameter families of embeddings

M ×∆k N ×∆k

∆k.

Math 272x Notes 23

There is also a sub-simplicial set SEmbc,∂,U (M,N) consisting of compactlysupported embeddings that is equal to some fixed g0 near ∂M ×∆k.

Corollary 7.6. The map SDiffc,∂,U (N)→ SEmbc,∂,U (M,N) is a Kan fibration.

Theorem 7.7. The map SEmbc,∂,U (M,N) → Sing(Embc,∂,U (M,N)) is anequivalence.

You can move the boundary as well as long as

(i) ∂M stays always in ∂N , or

(ii) ∂M stays always in int(N).

It is bad if ∂M touches ∂N at one point but lies in int(N) at another point.

Math 272x Notes 24

8 September 18, 2017

Last time we proved the parametrized isotopy extension on our way to diffeo-morphisms of S2. Next time I will do Gramain/Cerf’s proof of Smale’s theorem.We will be using parametrized isotopy extension throughout the course.

8.1 Embeddings of Rm

Theorem 8.1. The inclusions

O(m) → GLm(R) → Diff(Rm) → Emb(Rm,Rm)

are weak equivalences.

Proof. We’ll start with GLm(R) → Emb(Rm,Rm). There will be three steps:

(i) translate to fix the origin,

(ii) make it linear near the origin,

(iii) push non-linearity out to ∞.

The reason (iii) works is because we are using the weak Whitney topology, notthe strong one.

First, translation gives a deformation retract of Emb(Rm,Rm) to the sub-space of embeddings fixing the origin. Explicitly, we use

[0, 1] 3 τ 7→ (p 7→ h(p)− τh(0)).

In the second step, we need to show that each commutative diagram

Si GLm(R)

Di+1 Emb0(Rm,Rm)

may be homotoped through commutative diagrams such that there is a lift. Letη : Rm → [0, 1] be C∞ which is 1 near 0 and supported in Dm. By compactnessof Di+1, there exists an ε0 > 0 such that

[0, 1] 3 τ 7→(p 7→

(1− τη

( pε0

))Hs(p) + τη

( pε0

)D0Hs(p)

).

This is an embedding for all s ∈ Di+1 and τ ∈ [0, 1]. At τ = 0, it is Hs and atτ = 1, it is a linear map D0Hs near the origin.

For the third step, for λ ∈ (0,∞) let rλ be the diffeomorphism p 7→ λp.

Denote the result of the second step at τ = 1 by H(1)s . Consider

[0, 1] 3 τ 7→

r−11−τ H

(1)s r1−τ τ < 1

D0S(1)s τ = 1.

Math 272x Notes 25

This is continuous at τ = 0 is H(1)s and at τ = 1 is linear. Also, note that it

fixes GLm(R) pointwise.So we get that GLm(R) → Emb(Rm,Rm) is a weak equivalence.

O(m) GLm(R) Emb(Rm,Rm)

Diff(Rm)

' '

'

The map O(m) → GLm(R) is an equivalence by Gram–Schmidt, and GLm(R) →Diff(R) being an equivalence follows from that if we started out with a diffeo-morphism it would stay a diffeomorphism.

Note that the map Emb(Rm,Rm)→ GLm(R) given by taking the derivativeis a weak homotopy inverse.

Now let us look at embeddings into a manifoldM . The map taking derivativegeneralizes to

Emb(Rm,M)→ FrGL(TM); f 7→ (f(0), [D0f(~e1), . . . , D0f(~em)]).

Theorem 8.2. The map Emb(Rm,M)→ FrGL(TM) is a weak equivalence.

Proof. Consider the diagram

Emb(Rm,M) FrGL(TM)

M M.

ev0 π

id

Here the right vertical map is a locally trivial bundle and so a fibration, andthe left vertical map is also a fibration by parametrized isotopy extension. Soit suffices to prove that the map of fibers is a weak equivalence.

This map isEmb0(Rm,M)→ GLm(R),

but by shrinking Rm, we can modify any compact family in Emb07→p(Rm,M)to land it a chart. This reduces the problem to the case M = Rm.

8.2 Connected sums

Let M and N be oriented m-dimensional manifolds, and let ϕM : Dm →M bean orientation-preserving map, and ϕN : Dm → N be an orientation-reversingmap.

Definition 8.3. The connected sum is given by

M#N = (M \ int(ϕM (Dm))) ∪Sm−1 (N \ int(ϕN (Dm))).

Math 272x Notes 26

Note that this new manifold is oriented. We’ll show that this construction isindependent up to diffeomorphisms of ϕM and ϕN . (Here, we have to assumethat M and N are connected.)

Lemma 8.4. If ϕ′M is isotopic to ϕM , then

M#(ϕM ,ϕN )N ∼= M#(ϕ′M ,ϕN )N.

Proof. By isotopy extension, there is a diffeomorphism f : M → M such thatf ϕM = ϕ′M . Then we can construct

F : M#(ϕM ,ϕN )N →M#(ϕ′M ,ϕN )N ; p 7→

f(p) p ∈M \ int(ϕM (Dm))

p otherwise.

This is a diffeomorphism.

Thus to show that connected sums are well-defined up to diffeomorphism,it suffices to show that the space of orientation-preserving embeddings is path-connected.

Lemma 8.5. If M is path-connected and oriented, then Emb+(Dm,M) is path-connected.

Proof. We get a map

Emb+(Dm,M)→ Emb+(Rm,M)

by restricting to the interior. This is a homotopy equivalence because Rm → Dm

has an inverse up to isotopy. Now we have computed Emb+(Rm,M), which isthe oriented frame bundle FrGL,+(TM). There is then a fiber sequence

SO(n)→ FrGL,+(TM)→M.

Because the first and last are path-connected, the total space is also path-connected.

8.3 Diffeomorphisms of S2

Theorem 8.6. Diff(S2) ' O(3)

Proof. We have a fibration

Diff(S2)→ Emb(D2, S2)

by isotopy extension. Now the homotopy of the base can be computed as

Emb(D2, S2) ' Emb(R2, S2) ' FrGL(TS2) ' FrO(TS2) ' O(3).

The fiber of the map is those diffeomorphisms of S2 that fix D2. Then this isDiff∂,D(D2) ' ∗. There is a map O(3) → Diff(S2), and if you trace through allthe maps,

O(3) → Diff(S2)→ Emb(D2, S2) ' O(3)

is the identity.

Math 272x Notes 27

9 September 20, 2017

Today we will finish talking about 2-dimensional stuff. We will reprove Smale’stheorem Diff∂(D2) ' ∗.

9.1 Gramain’s proof of Diff∂(D2) ' ∗We will give a reformulation of Smale’s theorem in terms of spaces of arcs. Forthe diameter γ0, there is a action

Diff∂(D2)action on γ0−−−−−−−−→ Emb∂(I,D2).

This is a fibration by parametrized isotopy extension, and its fiber is going to beDiff∂(D2) × Diff∂(D2). Actually we need to be careful and sometimes replacethings with fixing the neighborhood of boundary and replacing by a homotopyequivalent thing, but this is annoying and we are not going to do this.

The long exact sequence of homotopy groups based at id is going to be

π2(Diff∂(D2))2 π2(Diff∂(D2)) π2(Emb[γ0],∂(I,D2))

π1(Diff∂(D2))2 π1(Diff∂(D2)) π1(Emb[γ0],∂(I,D2))

[g], [g−1] [id] [γ0].

Note that there is a section (up to homotopy) of Diff∂(D2)2 → Diff∂(D2) givenby g 7→ (g, id). This shows that the maps πk(Diff∂(D2))2 → πk(Diff∂(D2)) issurjective, and so we have short exact sequences. Then

πi+1(Emb[γ0],∂(I,D2)) ∼= πi(Diff∂(D2)).

So Smale’s theorem is equivalent to that Emb[γ0],∂(I,D2) is weakly contractible.

Extend the disk D2 to D2 on one side of γ0, and define T = D \ S. Thenthere is a fiber sequence

Emb∂(I ∪ S, D2relS)→ Emb∂(I ∪ S, D2)→ Emb(S, D2)

by isotopy extension. The last one is weakly equivalent to O(2) because it is anembedding of a disk into a manifold. Next, Emb∂(I ∪ S, D2) ' ∗ because wecan contract this lollipop to near the fixed point. So we can say that

Emb[γ0],∂(I, T ) = Emb[γ0],∂(I ∪ S, D2) ' ∗.

Now let β0 be the path on the handle part of T . Then clearly

Emb[γ0],∂(I, T \ β0) ' Emb[γ0],∂(I,D2).

Math 272x Notes 28

To show that this is weakly contractible, it suffices to show that the map

Emb[γ0],∂(I, T \ β0) → Emb[γ0],∂(I, T ) ' ∗

induces injections on π∗.Consider a universal cover T of T and identify this with R × I. Then

Emb[γ0],∂(I, T ) can be identified with the subspace E ⊆ Emb[γ0],∂(I,R × I)consisting of path that stays an embedding on T . Similarly Emb[γ0],∂(I, T \ β0)may be identified with Emb[γ0],∂(I, (−1, 1)× I).

Now we have a diagram commuting up to homotopy on compact support:

Emb[γ0],∂(I, T \ β0) Emb[γ0],∂(I, T )

E

Emb[γ0],∂(I, (−1, 1)× I) Emb[γ0],∂(I,R× I)

'

∼=

r

This shows that the first horizontal map induces injection on π∗. Therefore weget Emb[γ0],∂(I,D2) ' ∗.

γ0

β0

S

T

D2

Figure 1: Extending D2 to D2

9.2 Space of arcs

We could replace D2 by Σ a compact manifold. The path could go around somehandles in Σ and end at the same boundary component, and we could end at adifferent boundary component.

Theorem 9.1. Emb[γ0],∂(I,Σ) ' ∗.

Proof. Let’s first consider the first case, where the two paths have endpoints indifferent boundary components. Fill in one of a boundary component, and call

Math 272x Notes 29

this S. (We are taking Σ to be something like T .) Now we can do the sameargument and use the fiber sequence

Emb∂(I,Σ)→ Emb∂(I ∪ S,Σ ∪ S)→ Emb(S,Σ ∪ S).

The middle one is contractible by the lollipop thing, and the embedding of diskis going to be given by a frame bundle FrO(T (Σ ∪ S)). This lies in a fibersequence

O(2)→ FrO(T (Σ ∪ S))→ Σ ∪ S.

This shows that Emb(S,Σ ∪ S) has no πi for i > 1.For the second case when the endpoints of the path lie in the same boundary

component, we attach a handle as we’ve done for D2 and form T . Take a pathβ0 inside the handle. It again suffices to show that

Emb[γ0](I, T \ β0)→ Emb[γ0],∂(I, T )

is injective on homotopy groups.We use a cover T of T corresponding to π1(Σ) ⊆ π1(T ) = Z∗π1(Σ). We can

use this to get a retraction of Emb[γ0],∂(I, U) to Emb[γ0],∂(I, T \ β0).

9.3 Path components of Diff∂(Σ)

Theorem 9.2. If Σ is an oriented path-connected compact surface with ∂Σ 6= ∅,then the path-components of Diff∂(Σ) are weakly contractible.

Proof. It suffices to prove this for the path component Diff id∂ (Σ) of the identity.

We induct over the genus g and the number of boundary components n. Thecase (g, n) = (0, 1) is D2 and Smale’s theorem.

Pick a path γ0, and then there is a fibration

Diff id∂ (Σg,n rel γ0)→ Diff id

∂ (Σg,n)→ Emb[γ0],∂(I,Σg,n).

Then this Diff id∂ (Σg,n rel γ0) is actually Diff id

∂ (Σg,n−1) or Diff id∂ (Σg−1,n+1).

Math 272x Notes 30

10 September 22, 2017

Hatcher proved Diff∂(D3) ' ∗, and we are going to prove an almost equivalentstatement about embeddings.

10.1 Restating Smale’s theorem

Proposition 10.1 (Restatement of Smale’s theorem). The map Emb(D2,R2)→Emb(S1,R2) is a weak equivalence.

Proof. We have

Diff(R2) Diff(R2)

Emb(D,R2) Emb(S1,R2),

where the vertical maps are fibrations by parametrized isotopy extension. Themap of fibers is

Diff(R2 relD2)→ Diff(R2 relS1) ∼= Diff(R2 relD2)×Diff∂(D2).

Now Diff∂(D2) is contractible by Smale’s theorem, and so it is a weak equiva-lence. Now we can use the long exact sequence.

We still have to show that Emb(D2,R2) → Emb(S1,R2) is surjective onπ0.

Proposition 10.2. Every smooth embedding ϕ : S1 → R2 extends to a smoothembedding D2 → R2.

Proof. I’m only going to give a sketch. We’ll use from Morse theory that everysmooth map S1 → R can be perturbed to have the following properties:

(i) the set of p ∈ S1 such that Dpf = 0 is finite,

(ii) the f(p) ∈ R for p critical point are distinct,

(iii) near each critical point p we can find coordinates x on S1 such that pcorresponds to x = 0 and f(x) = f(p) + αx2 for α 6= 0.

Let us considerh : S1 ϕ−→ R2 π2−→ R.

and perturb h to h that satisfies (i), (ii), (iii). Since embeddings are open inC∞(S1,R2), linear interpolation from ϕ to ϕ with y-coordinate replaced by h.

By isotopy extension, ϕ can be extended to D2 if and only if ϕ can be. Nowtake a set yiki=1 ⊆ R such that the lines R× yi separate all critical points.By linear interpolation, we can make im(ϕ) vertical near R × yi. Cut alongR× yi by innermost pairs (points in im(ϕ) ∩ (R× yi) without intersectionpoints between them).

This cuts the map S1 → R into many embeddings S1 → R. Each ofthese individual pieces can be isotoped to have one of the five local models:

Math 272x Notes 31

a rectangle, a parabola bounded above, parabola bounded below, and verticalreflections. Then we have formulas for disks that bound these local models, andisotopy extension gives that each of these pieces actually bound disks.

Now we need to reverse the cuts by adding or subtracting disks.

Lemma 10.3. If M is a smooth manifold and ϕ0 : Dm−1 → ∂Dm and ϕ1 :Dm−1 → ∂M are embeddings, then M ∼= M ∪Dn−1 Dn.

Using this lemma, we can glue these maps together to get a extension of ϕ toD2. The parametrization of im(ϕ) and ϕ on ∂D2 might differ, but π0 Diff(S1) =Z/2Z so we can line them up well.

10.2 Hatcher’s theorem

Theorem 10.4 (Hatcher). Emb(D3,R3)→ Emb(S2,R3) is a weak equivalence.

Note that this is equivalent to Diff∂(D3) ' ∗ and that every embedded S2

in R3 bounds a disk.

Proposition 10.5 (Alexander’s theorem). Every S2 → R3 bounds a disk.

Proof. First, make a height function that generic Morse. Next pick non-criticalziki=1 such that the planes R2×zi separate all the critical points. Cut alongR2×zi inner most circles first. Isotope pieces to be given by one of 7 standardmodels using Emb(S1,R2). These are all given by formulas, and we can checkthat they all bound D3. Now reverse the isotopies and reassemble to D3 usingthe lemma about adding and subtracting disks. Finally use Diff(S2) ' O(3) toadjust parametrization of ∂D3.

Why can’t you do this in dimension 4? In this case, the standard modelswill contain some handle bodies. Then the lemma about adding and subtractingdisks is not going to work. It is really something special about dimension 3.

Corollary 10.6. Diff(S3) ' O(4).

Corollary 10.7 (Hatcher–Ivanov). If M is a Haken 3-manifold, then

Diff id∂ (M)→ hAutid

∂ (M)

is a weak equivalence.

Math 272x Notes 32

11 September 25, 2017

We have been comparing diffeomorphism groups, did some dimension 2 and alittle bit of dimension 3. We are now going towards proving the s-cobordismtheorem and using this to understand πiM∂(Dn) for n ≥ 6 and i = 0, 1.

In this lecture, all manifolds have empty boundary.

11.1 Sard’s lemma

Definition 11.1. Let f : M → N be a smooth map. A point a ∈M is said tobe a critical point if Dqf : TqM → Tf(q)N is not surjective. Then f(q) ∈ Nis said to be a critical value.

If Dqf : TqM → Tf(q)N is surjective, then q is said to be a regular point.A point p ∈ N is a regular value f−1(p) consists of regular points, i.e., p isnot a critical value.

The inverse function theorem says that f−1(M) ⊆ N is a smooth manifoldif p ∈ N is a regular value (and f is proper).

Theorem 11.2 (Sard’s lemma). Let U ⊆ Rm be an open subset and let f :U → Rn be a smooth map. Then the set Crit(f) of critical values has measurezero.

Since measure zero subsets cannot contain subsets, the set of regular valuesis dense.

Corollary 11.3. The set Crit(f) for f : M → N smooth has measure zero.

Proof. Let us first do the case when M = Ui ⊆ Rm is open. Then cover N bya countable collection of charts. ϕi : N ⊇ Vi →Wi ⊆ Rn. Then

Crit(f) =⋃i

ϕ−1i (Crit(ϕi f |f−1(Vi)))

and so is a countable union of measure zero sets. Hence it has measure 0.For the general case, cover M by charts and do the same thing. Then the

critical values are a countable union of measure 0 sets, and so is also of measure0.

Proposition 11.4. Every compact smooth M can be embedded in R2m+1.

Proof. Take some embedding ϕ : M → RN . If N ≤ 2m + 1, we are done.Otherwise, we will show how to reduce N by 1. The strategy is to consider theprojection

πv⊥ ϕ : M → RN−1

where πv⊥ is the projection onto the hyperplane v⊥. We’ll find a v such thatthis is still an embedding.

Two things can go wrong.

Math 272x Notes 33

(i) The map might not be injective, or

(ii) the derivative D(πv⊥ ϕ) might not be injective somewhere.

(i) happens if v lies in the image of

F : M ×M \∆M → SN−1; (x, y) 7→ ϕ(x)− ϕ(y)

‖ϕ(x)− ϕ(y)‖,

and (ii) happens if v lies in the image of

G : S(TM) = w : ‖w‖ = 1 → SN−1; w 7→ Dϕ(w)

‖Dϕ(w)‖.

If N > 2m + 1, then im(F ) = Crit(F ) and im(G) = Crit(G) because thedifferential can never be surjective by counting the dimension. (The codomainhas dimension ≥ 2m+ 1 but the domain has dimension 2m.) So there exists avector v ∈ SN−1 \ (im(F ) ∪ im(G)) by Sard’s lemma.

11.2 Transversality

Definition 11.5. Let f : M → N and g : M ′ → N be smooth maps. We saythat f is transverse to g (or f t g) if for all p ∈ f(M) ∩ g(M ′), q ∈ f−1(p),q′ ∈ g−1(p), we have that

Dqf(TqM) +Dqg(TqM′) = TpN.

Example 11.6. If m+m′ < n, then f t g if and only if the images are disjoint.

There are special cases:

(1) If g is an inclusion of a submanifold X, then we use f t X instead off t g, and this is equivalent to for all q ∈ f−1(f(M) ∩X), the map

TqMDqf−−−→ Tf(q)N

N−→ (νX)q

is surjective. In this case, the implicit function theorem says that f−1(X)is a smooth submanifold. In particular, f t x is equivalent to that x isa regular value of f .

(2) If f is also an inclusion of a submanifold M , then we use M t X insteadof f t X. This is equivalent to TpM+TpX = TpN for all p ∈M ∩X. Theimplicit function theorem then implies that there exists a chart V aroundp in which M and X look like generically intersecting affine planes.

Theorem 11.7. Every smooth map f : M → N can be approximated arbitrarilyclosely by a smooth map transverse to X.

Proof. We shall prove a strongly relative version. Consider closed sets Cdone

and Dtodo and open set Udone ⊇ Cdone and Vtodo ⊇ Dtodo. Suppose f is already

Math 272x Notes 34

transverse to X on an Udone and we need to make it transverse to X near Dtodo

without modifying it on a neighborhood on

Cdone ∪ (M \ Vtodo).

(Here U and V need not be disjoint.)First, consider the case when M = U ⊆ Rm is open and X = 0 and

N = Rr. The condition f t 0 is equivalent to 0 being a regular value of f .By Sard’s lemma, regular values are dense in Rr and so we can pick a sequenceof regular values xn such that xn → 0. Consider fk = f − xk. This will messwith everything, so pick a bump function

η : U → [0, 1]

such that η is 0 near Cdone ∪ (M \ Vtodo) and 1 near Dtodo intersect a smallerneighborhood of Cdone ⊆ Udone. Instead, take fk = f − ηxk. The claim is thatif xk is small enough, then xk is transverse to 0 both near Cdone ∪ Dtodo. Wewill continue next time.

Math 272x Notes 35

12 September 27, 2017

Last time I made this claim

Theorem 12.1. Every smooth map f : M → N (with ∂M = ∂N = ∅) can beapproximated by a smooth map transverse to X.

Proof. We are going to prove the strongly relative version. Let f : M → Nsuch that f t X on Udone ⊇ Cdone. We are going to fix the function near Dtodo

without changing it near Cdone ∪ (M \ Vtodo).

Figure 2: Figure of the strongly relative version setup

Case I. Suppose that M ⊆ U = R is open, X = 0, N = Rr and Dtodo iscompact. The idea was to consider f −xk where xk are regular values of f thatconverge to 0. Take η : M → [0, 1] smooth, such that η = 1 near Dtodo \ Utodo,and η = 0 near Cdone ∪ (M \ Vdone).

Let us take the function f − xkη. This is transverse to 0 near Cdone,because it is f , and near Dtodo \ Udone because there it is f − xk. What aboutDtodo ∩ Udone? For a compact subset K, The condition of being transverse onK is an open condition, and so if k is large, the function is transverse.

Case II. Suppose M = U ⊆ Rm is open, Dtodo is compact, and N = X×Rr.Then f t X is equivalent to π f t 0. So it reduces to Case I.

Case III. Suppose M = U ⊆ Rm is open, Dtodo is compact, νX is trivializ-able. Then the tubular neighborhood theorem gives an embedding X×Rr → N .Now we substitute

M ′ = f−1(X × Rr), f ′ = f |M ′ ,C ′done = Cdone ∩M ′, U ′done = Udone ∩M ′,D′todo = f−1(X ×Dr), V ′todo = Vtodo ∩ f−1(X × int(2Dr)).

Then it reduces to Case II.

Math 272x Notes 36

Case IV. In the general case, we induct over charts. There exists a locallyfinite covering of X, Uα such that νX |Uα are all trivializable. Then there existsa locally finite chart of M by charts ϕi : M ⊇ Vi →Wi ⊆ Rm such that

(i) 2Dm ⊆Wi,

(ii) Dtodo ⊆⋃i ϕ−1i (Dm),

(iii) for all i, there exists α such that f(Vi) ∩X ⊆ Uα.

We can order the i and identify it with N.By induction we construction fi : M → N which is transverse to X on Ui of

Ci = Cdone ∪⋃j≤i

ϕ−1i (Dm).

For the induction step, apply Case II to

M ′ = Wi+1, f = fi ϕ−1i+1

C ′done = ϕi+1(Ci ∩ Vi+1), U ′done = ϕi+1(Ui ∩ Vi+1),

D′todo = ϕi+1(Dtodo ∩ Vi+1) ∩Dm, V ′todo = ϕi+1(Vtodo ∩ Vi+1) ∩ int(2Dm).

after this, you take f = limi→∞ fi, which is smooth and well-defined becausenear each point you change only finitely many times. Then f t X near Cdone ∪⋃i ϕ−1i (Dm).

By applying this inclusion of a submanifold,

Corollary 12.2. If M,X ⊆ N are compact submanifolds, then there exists anarbitrary small ambient isotopy ϕt on N such that ϕ1(M) t X.

12.1 Jet transversality

We can reprove the previous theorem using jet transversality. Recall thatJ1(M,N) is given by pairs (m, [g]) of m ∈ M and a 1-jet of smooth mapsg : M → N near m. This 1-jet encodes the value of g and the first derivative.This is a smooth manifold of dimension m+ n+mn. It is a locally trivial fiberbundle over M ×N with fiber a finite-dimensional vector space. There is a map

j1 : C∞(M,N)→ Γ(M,J1(M,N)); f 7→ (m 7→ (m, [f ]m)).

By considering local coordinates, we see that the images are smooth sections.Let X ⊆M be a submanifold, and define X ⊆ J1(M,N) as the submanifold

consisting of (m, [g]) for m ∈ X.

Proposition 12.3. f t X if and only if j1(f) t X .

The transversality result then also follows form the stronger theorem.

Theorem 12.4 (Jet transverality). Let D ⊆ Jr(M,N) be a stratified subset.Then every f : M → N can be approximated f such that jr(f) t D.

Math 272x Notes 37

Here, a subset Y of a manifold is stratified if it is a finite union of i-dimensional manifolds Yi, with Yi∩Yj = ∅ if i 6= j and Y i =

⋃j≤i Yj . Transver-

sality then is transversality to all the strata.Morse functions are function f : M → R with non-degenerate critical points,

and this can be rephrased as j2(f) t DMorse.

Example 12.5. Let M,N be smooth manifolds and suppose m > n. Thenf : M → N will have a dense set of regular values in N , but the rank of thederivative drops somewhere for algebraic topology reasons. How nice can wemake the set where the derivative has i-dimensional cokernel?

Let Σi ⊆ J1(M,N) be the subspace of (m, [g]) where the cokernel of thederivative part of [g] has dimension i. Then Σi is a stratified subset (coveringJ1(M,N)) and the codimension of Σi is i(m−n+ i). Jet transversality impliesthat we can approximate f by f such that Σi(f) = m : dim cokerD = i is acodimension i(m− n+ i) smooth manifold.

Math 272x Notes 38

13 September 29, 2017

We will use jet transversality and develop Morse theory.

13.1 Morse functions

Definition 13.1. Morse functions are smooth functions f : M → R with“nondegenerate critical points”.

Given a function f : Rm → R, its Hessian H0(f) at 0 is defined in terms ofthe Taylor approximation

f(x) = f(0) + 〈∇0f, x〉+1

2〈H0(f)x, x〉+ · · · .

The problem is that the Hessian is not invariant under coordinate change. How-ever, it turns out that when 0 is a critical point, it is well-defined.

For a critical point p ∈ M of f : M → R, we can consider for v, w ∈ TpMand extend it to a vector field v, w near p. Then

v(w(f))[p]− w(v(f))[p] = [v, w](f)[p] = 0

because p is a critical point. Also the left hand side is independent of theextension v and the right hand side is independent of the extension w. Hencethis expression is independent of neither and we get a well-defined function

Hessp(f) : TpM ⊗ TpM → R.

Definition 13.2. A critical point p ∈M of f is nondegenerate if Hessp(f) isnondegenerate.

Classification of symmetric bilinear forms over R implies that there is a basisx1, . . . , xm of TpM such that Hessp(f) in this basis is −

∑λi=1 x

2i +

∑mi=λ+1 x

2i .

This λ is called the index of the critical point.

Definition 13.3. A function f : M → R is called Morse if all critical pointsare non-degenerate and f is regular near ∂M .

Existence of Morse functions can be proved using jet transversality. Considera subset

DnotMorse ⊆ J2(M,R)

of 2-jets (m, [g]) with [g] has vanishing first derivative at m and degenerate Hes-sian at m. In terms of local coordinates, J2(Rm,R) correspond to polynomialsin m variables of degree ≤ 2 and no constant. So this is the subset of∑

i,j

hijxixj

Math 272x Notes 39

where hij is symmetric and non-degenerate. This can be stratified by the di-mension of the kernel of the Hessian. From the local description, we see that itis a stratified subset in the sense of Whitney.

One immediate observation is that f is Morse if and only if j2(f)∩DnotMorse =∅. But note that codim(DnotMorse) is m+ 1. This means that

j2(f) ∩ DnotMorse = ∅ ⇐⇒ j2(f) t DnotMorse.

Apply jet transversality relative to the boundary.

Theorem 13.4. If f : M → R is a smooth and regular near ∂M , then f canbe approximated by f which is Morse and f = f near ∂M .

Corollary 13.5. Morse functions exist.

13.2 Morse lemma

Definition 13.6. A critical point p ∈ M of a smooth function f : M → R issaid to be a Morse singularity if there exists a chart around p with coordinates(x1, . . . , xm) (p corresponds to 0) such that

f(x1, . . . , xm) = f(0)−λ∑i=1

x2i +

m∑i=λ+1

x2i .

An easy computation shows that a Morse singularity is a nondegeneratecritical point. The Morse lemma shows that the converse is true.

Lemma 13.7 (Morse lemma). Every nondegenerate critical point is a Morsesingularity.

Proof. Without loss of generality, assume f(p) = 0. By taking a chart, we mayassume M = W ⊆ Rm and p = 0. In these conditions, Taylor approximationsays that there is a smooth map Q : W → Sym(Rm) (symmetric matrices) suchthat

f(x) = 〈Q(x)x, x〉

and Q(0) = Hess0(f).We would like to find coordinates y such that Q is independent of y. Make

an ansatz y = A(x)x for a smooth map A : W → GLm(R) with A(0) = 1. Weneed to solve the equation

〈Q(x)x, x〉 = 〈Q(0)A(x)x,A(x)x〉,

or equivalently, A(x)tQ(0)A(x) = Q(x). To do so, consider

G : Sym(Rn)×W → Sym(Rn);

(B, x) 7→ (id + 12Q(0)−1B)tQ(0)(id + 1

2Q(0)−1B)−Q(x).

Math 272x Notes 40

This map is smooth, G(0, 0) = 0, and its derivative with respect to B at B = 0is

( 12Q(0)−1)tQ(0) +Q(0) 1

2Q(0)−1 = 1.

So by the inverse function theorem, there is a β : U → Sym(Rm) such thatG(β(x), x) = 0. Thus A(x) = id + 1

2Q(0)−1β(x) solves the equation.Now we have coordinates y such that

f(y) = 〈Q(0)y, y〉.

Apply some linear coordinate change to put the symmetric matrix Q(0) in di-agonal form with entries ±1.

Note by the Morse lemma, non-degenerate critical points are isolated andthus if M is compact, a Morse function f : M → R can only have finitely manycritical points.

Math 272x Notes 41

14 October 2, 2017

Last time we proved existence of Morse functions f : M → R with non-degenerate critical points, and we showed that near a critical point, there existcoordinates such that f takes the form f = c−

∑λi=1 x

2i +

∑mi=λ+1 x

2i .

14.1 Generic Morse function

Definition 14.1. A Morse function f : M → R is generic if all critical valuesare distinct.

Lemma 14.2. Any Morse function can be arbitrarily approximated by genericMorse functions.

Proof. Let f : M → R be a Morse function with critical points pi ⊆ M ,and for each pi pick ηi : M → R which are compactly supported and 1 near p.Consider

f +∑i

εiηi

for small εi ∈ R. This is a Morse function with same critical points if εi is smallenough. Now take εi so that the critical points take different values.

14.2 Understanding level sets

If we have a function f : M → R, then the subsets f−1(a) ⊆M are called levelsets, and is a submanifold if a is a regular value of f . We’ll try to explain whatf−1([a, b]) looks like in two cases:

I. f−1([a, b]) contains no critical points.

II. f−1([a, b]) contains a unique critical point p, with f(p) 6= a, b.

These are enough to understand f generic Morse.

Proposition 14.3. If f : M → R is proper and f−1([a, b]) contains no criticalpoint, then f−1([a, b]) ∼= f−1(a)× [a, b] and f−1(b) ∼= f−1(a).

Proof. We;ll use a smooth vector field χ on f−1([a, b]) such that df(χ) = 1.(This can be done by looking at a partition of unity or a Riemannian metric.)Because f is proper, f−1([a, b]) is compact and the flow Φx exists until flowlineshit the boundary. Because df(χ) = 1, if we start at p ∈ f−1(a) then we hitf−1(b) at time b− a. The diffeomorphism is going to be

f−1(a)× [a, b]→ f−1([a, b]); (p, t) 7→ Φχ(p, t− a).

Now let’s look at the second case, when there is a unique critical point pin the inverse image f−1([a, b]). Denote c = f(p), and fix some nice Morse

Math 272x Notes 42

singularity coordinates (x1, . . . , xn) around p, defined on W ⊆ Rm. Now we canwrite

f(x) = c−λ∑i=1

x2i +

m∑i=λ+1

x2i .

Take an ε > 0 such that B√2ε(0) ⊆W and a < c− 2ε < c+ 2ε < b.

The claim is that f−1([a, c+ ε]) deformation retracts onto

Rf−1([a, c− ε]) ∪ C,

where

C =

(x1, . . . , xλ, 0, . . . , 0) :

λ∑i=1

x2i ≤ ε

.

The strategy to find a manifold with boundary U , obtained from f−1([a, c− ε])by a “handle attachment, such that the inclusions f−1([a, c − ε]) ∪ C → U →f−1([a, b]) are deformation retracts.

We modify f to F as follows. Pick a smooth ϕ : [0,∞) → [0,∞) such that(i) ϕ(0) ∈ (ε, 2ε), (ii) ϕ(t) = 0 for t ∈ [ε,∞), and (iii) ϕ′(t) ∈ (−1, 0]. Thendefine

F : M → R; x 7→

f(x)− ϕ(

∑λi=1 x

2i + 2

∑mi=λ+1 x

2i ) x ∈ V

f(x) otherwise.

Lemma 14.4. F has the following properties:

(i) f−1([a, c+ ε]) = F−1([a, c+ ε]).

(ii) F has the same critical points as f .

(iii) F−1([a, c−ε]) = f−1([a, c−ε])∪(Dλ×Dn−λ), attached along ∂Dλ×Dm−λ.

Proof. (i) Because F ≤ f , we have f−1([a, c+ ε]) ⊆ F−1([a, c+ ε]). Conversely,

let x ∈ F−1([a, c + ε]) such that ϕ(∑λi=1 x

2i + 2

∑mi=λ+1 x

2i ) > 0. Let’s write

x = (y, z) ∈ Rλ × Rm−λ so that this is ‖y‖2 + 2‖z‖2. Then ϕ(t) = 0 for t > εimplies ‖y‖+ 2‖z‖ ≤ ε. Then

f(x) = c− ‖y‖2 + ‖z‖2 ≤ c+1

2‖y‖2 + ‖z‖2 ≤ c+

1

2ε.

This proves F−1([a, c+ ε]) ⊆ f−1([a, c+ ε]).(ii) We only need to check this if x ∈ V . We explicitly can check

1

2∇F (x) = (−y − ϕ′(‖y‖2 + 2‖z‖2)y, z − ϕ′(‖y‖2 + 2‖z‖2)2z).

Since −1 < ϕ′(t) ≤ 0, we have this equal to zero if and only if y = 0 and z = 0.(iii) This is annoying to prove. Read Milnor’s book.

We have some conclusion. We have f−1([a, b]) ∼= U ∪ F−1(c+ ε)× [c+ ε, b].This follows from Case I, because there are no critical values in [c + ε, b]. Wethen have

f−1([a, b]) ∼= (f−1(a)× [a, c− ε]) ∪ (Dλ ×Dm−λ).

Math 272x Notes 43

14.3 Handle decomposition

Definition 14.5. If M is a smooth manifold with boundary, and ϕ : ∂Di ×Dm−i → ∂M is a smooth embedding, then

M ∪ϕ (Di ×Dm−i)

is a smooth manifold with boundary after smoothing corners. This is said to beobtained from M by attaching handles.

Definition 14.6. A handle decomposition of a smooth manifold M is a wayof writing M as an iterative handle attachments starting at ∅.

Combining existence of generic Morse functions with the link between criticalpoints and handle attachments, you could prove the following.

Corollary 14.7. Every compact M admits a finite handle decomposition.

Math 272x Notes 44

15 October 4, 2017

Last time we showed that M has a handle decomposition.

15.1 Cobordism

Definition 15.1. Given closed m-dimensional manifolds M0 and M1, a cobor-dism (W,∂0W,∂1W, f0, f1) from M0 to M1 is an m + 1-dimensional mani-fold W with ∂W = ∂0W q ∂1W with diffeomorphisms f0 : M0 → ∂0W andf1 : M1 → ∂1W .

Example 15.2. There is a cobordism between two circles and three circles. IfW is a closed (m+ 1)-dimensional manifold, then this is a cobordism ∅ → ∅.

Handle attachments give rise to cobordism as follows. If

ϕ : ∂Di ×Dw−i →M0 × 1

is a smooth embedding, where w = m+ 1, the new manifold

(M0 × I) + (ϕ) = (M0 × I) ∪ϕ (Di ×Dw−i)

is a cobordism from M0 to

M1 = (M0 \ ϕ(∂Di × int(Dw−i))) ∪ (Di × ∂Dw−i),

which is the “result of surgery of M0 along ϕ”.It turns out that all cobordisms are handle attachements. Using the relative

version of existence of generic Morse functions, one proves the following.

Proposition 15.3. Every cobordism W is diffeomorphic relative to ∂0W to

(∂W0 × I) + (ϕ1) + · · ·+ (ϕk).

(Note that the order is important; it might not even make sense.) In particular,∂1W is diffeomorphic to ∂0W modified by finitely many surgeries.

We’ll develop tools to manipulate such handle decompositions. Today we willshow that we can change ϕi up to isotopy, and that if index(ϕi) ≥ index(ϕi+1)then we can interchange them after isotopy. Then we will show that if index(ϕi+1) =index(ϕi) + 1 and the attaching sphere of ϕi+1 intersects the transverse sphereof ϕi transversally in a single point, you can cancel both.

Let’s define some words. Suppose we are attaching a 1-handle D1 × D2.This i = 1 is called the index of the handle. Inside this handle there is a core

D1 × 0,

and its boundary ∂D1×0 is called the attaching sphere. Dually, there is acocore 0 ×D2 and a transverse sphere 0 × ∂D2.

Math 272x Notes 45

15.2 Handle isotopy, rearrangement, and cancellation

Lemma 15.4. If ϕ1 is isotopic to ϕ′1, then W+(ϕ1) is diffeomorphic to W+(ϕ′1)relative to ∂0W .

Proof. Let ϕti : ∂Di × Dw−i × [0, 1] → ∂1(W ) × [0, 1] denote the isotopy ofembeddings. Then there exists an isotopy of diffeomorphisms

f t : ∂1W × [0, 1]→ ∂1W × [0, 1]

such that f0 = id and ϕt1 = f t ϕ01, by isotopy extension. Take a collar

c : ∂1W × [0, 1)→W relative to ∂1W . Then the map

F : W + (ϕ1)→W + (ϕ′i); p 7→

c(f1−t(q), t) p = c(q, t)

p otherwise

is a diffeomorphism.

Lemma 15.5. Given a cobordism W + (ϕ0) + (ϕ1) with i0 = index(ϕ0) ≥ i1 =index(ϕ1), then we may isotope ϕ1 to ϕ′1 with image lying in

∂1W \ ϕ0(∂Di × int(Dw−i)) ⊆ ∂1(W + (ϕ0)).

Proof. First we make the attaching sphere of ϕ1 transverse to transverse sphereof ϕ0, using transversality results. But the dimension of the attaching sphereof ϕ1 is i1 − 1 and dimension of the transverse sphere of ϕ0 is w − i0 − 1. Butboth are submanifolds of ∂1W , which has dimension w − 1.

Next, we shrink in Dw−i1 to make the image of the attaching map ϕ1 disjointfrom the transverse sphere of ϕ0. Now we can flow the attaching sphere out ofthe handle (ϕ0). To do this, pick a vector field χ that points radially outwardsin the Dw−i0 \ 0 direction on (Di \ 0)× ∂Dw−i0 .

Once we have this, the image of ϕ′1 lies in the complement ∂1W \ϕ0(∂Di0×Dw−i0). Then we have

W + (ϕ0) + (ϕ′1) ∼= W + (ϕ′1) + (ϕ0)

relative to ∂0W . So by handle isotopy lemma, this is diffeomorphic to W +(ϕ0) + (ϕ1).

We can rearrange handles inductively to get the following.

Corollary 15.6. Every cobordism W is diffeomorphic relative ∂0W to

W = (∂0W × I) +∑

0-handles

(ϕ0i0) + · · ·+

∑w-handles

(ϕwiw).

This is equivalent to the existence of self-index Morse functions. This meansthat

f(∂0W ) = −1, f(∂1W ) = w + 1,

Math 272x Notes 46

and for each critical point p, f(p) is its index.Suppose we have a cobordism between 1-manifolds. If you attach a 0 sphere,

which is just a separate D2, and then you attach a 1-handle, which looks like aD1 ×D1 connecting it with the original cobordism, you just get

W ∪D1D2∼= W.

Lemma 15.7. Given W + (ϕ0) + (ϕ1) with index(ϕ1) = index(ϕ0) + 1 andattaching sphere of ϕ1 intersecting the transverse sphere of ϕ0 transversally ina single point, then W + (ϕ0) + (ϕ1) ∼= W relative to ∂0W .

Proof. This is going to be very hard to write in symbols and very hard to read.Firstly, what you do is use interpolation and make ϕ1 standard near the

intersection point. Next, by shrinking in the first Dw−i−1-direction and inter-polating, we can make im(ϕ1) standard near the intersection. Now we flowoutward ϕ1 along the outward vector field as before, until the intersection ofthe image of ϕ1 with (ϕ0) is standard. But now, we can use ϕ1 as coordinates(ϕ0 to reduce to the standard model. Now this is explicit enough to write downa diffeomorphism.

Math 272x Notes 47

16 October 6, 2017

Today I will continue talking about handle decompositions. Before doing thatwe need to talk about how these are related to the topology.

16.1 Handle decompositions and the topology

Let’s suppose that

W = (∂0W × I) +∑I0

(φi0)0 + · · ·+∑Iw

(φwiw),

and let Wk ⊆W be given by

Wk = (∂0W × I) +∑I0

(φ0i0) + · · ·+

∑Ik

(φkik).

Now W is an iterated pushout starting with W−1 = ∂0W × I∐Ik∂Dk ×Dw−k Wk1

∐IkDk ×Dw−k Wk.

qφkik

By collapsing Dw−k to ∗, we get a relative CW-complex X with weak homotopyequivalence f : W → X relative to ∂0W . We can do this iteratively by pushouts∐

Ik∂Dk Xk−1

∐IkDk Xk.

Here are some properties:

• h-cells correspond bijectively to h-handles.

• The degree of attaching map Dk+1ik+1

with respect to Dkik

is given by the

intersection number of attaching sphere of (φk+1ik+1

) with the transverse of

(φkik).

This allows you to pass information between topology and geometry.Here is one example. If you have a handle decomposition of W , we can read

this backwards. Instead of starting at ∂0W , we can start at ∂1W and theni-handles become (w− i)-handles. If W is oriented, we get we can use it to alignorientations of cores and the orientations of opposite direction cocores. Thenthe degree of the attaching maps will be the same. So we get, if W is oriented,then

H∗(W,∂0W ) ∼= H∗(X, ∂0W ) ∼= Hw−∗(Xrev, ∂1W ) ∼= Hw−∗(W,∂1W ).

Math 272x Notes 48

Definition 16.1. W is an h-cobordism if both maps ∂0W →W and ∂1W →W are homotopy equivalences.

16.2 Handle exchange

To trade an i-handle for an (i + 2)-handle, we will introduce a custom-builtpair of (i+ 1)-handles and (i+ 2)-handles such that the (i+ 1)-handle cancel toi-handle.

Lemma 16.2. Given an embedding Dw−1 → ∂1W , let ψi : ∂Di × Dw−i →∂1(W ) be given by restriction to a standard ∂Di ×Dw−i ⊆ Dw−1, there existsa ψi+1 : ∂Di+1 ×Dw−i−1 → ∂1(W + (ψi)) such that W ∼= W + (ψi) + (ψi+1).

Proof. This can be drawn so that attaching the two is just attaching Dw.

Let us write

∂1(Wk) = ∂1(Wk) \∐Ik+1

ϕk+1ik+1

(∂Dk+1 ×Dw−k−1).

Lemma 16.3 (Handle exchange). Suppose that for an i-handle φiji we have an

embedding ψi+1 : ∂Di+1 ×Dw−i−1 → ∂1(Wi) such that

(a) in ∂1(Wi) ⊇ ∂1(Wi), ψi+1 is isotopic to an embedding whose restriction to

∂Di+1 × 0 intersects the transverse sphere of (φiij ) once transversally,

(b) in ∂1(Wi+1) ⊇ ∂1(W1) the map ψi+1 is isotopic to an embedding which istrivial, i.e., is an restriction of an embedding Dw−1 → ∂1(Wi+1) to thestandard ∂Di+1 ×Dw−i+1.

Then there is an attaching map ψi+2 such that

W = · · ·+∑j 6=ij

(φij) +∑Ii+1

(φi+1ii+1

) + (ψi+2) + · · · .

Proof. Using triviality of ψi+1, we can build a ψi+2 so that it cancels with ψi+2.Then

W ∼= · · ·+∑Ii

(φiii) +∑Ii+1

(φiii+1) + (ψi+1) + (ψi+2)

∼=∑j 6=Ij

(φij) +∑Ii+1

(φi+1ii+1

) + (ψi+2).

16.3 Removing 0, 1, w − 1, w-handles

Lemma 16.4. If w ≥ 6 and ∂0W → W is 1-connected, then there exists ahandle decomposition with 0 or 1-handles.

Math 272x Notes 49

Proof. First pick a handle decomposition with handles appearing in order ofincreasing index. Now each 0-handle adds a new path component unless it islater reconnected by a 1-handle. This is because ∂0W →W is a bijection on π0.Also, the attaching sphere of this 1-handle automatically intersects the trans-verse sphere ∂Dw transversally in a single point. So we may use cancellation tocancel all the 0-handles.

Now we exchange 1-handles for 3-handles. We want to build a ψ2 : ∂D2 ×Dw−2 → ∂1(W1) which satisfies the conditions for handle exchange with respectto the chosen 1-handle (φ1

1). We build ψ2|∂D2 first. Consider the segmentS1

+ = D1×~e1 through the 1-handle. The endpoints are in the same component

because ∂0W →W is a π0-bijection. So we can connect them by S1− ⊆ ∂1(W0).

Note that

∂1(W0) = ∂1(W0) \∐

(∂D1 ×Dw−1) ∼= ∂0W \ (disks)

since there are no 0-handles. Because w ≥ 6, we have that π1(∂1(W0)) ∼=π1(∂0W ) π1(W ), we can assume that S1 = S1

+ q S1− is null-homotopic in W

by adding another loop if necessary.This gives an embedding (after perturbation) ψ2|S1

: S1 → ∂1(W1) thatpasses through the handle (φ1

1) around once. Because attaching spheres of 2-

handles are S1, we can also assume that the embedding is ψ2|S1: S1 → ∂1(W1).

Then by a small section of the normal bundle, we get ψ2 : ∂D2 → Dw−2 →∂1(W1).

The map ∂1(W2) →W is an isomorphism of π1, because to get from ∂1(W2)to W , we need to attach i-cells for i ≥ 3 and w − 2 cells and w − 1 cells on thebackwards direction. So we’re not changing π1 in this process. This means thatψ2 on S1 is null-homotopic in ∂1(W1) ⊆ ∂1(W2). That is, it bounds a disc in∂1(W2).

Math 272x Notes 50

17 October 11, 2017

I will prove the Whitney trick next time.

17.1 Two-index lemma

Recall that if a cobordism W is a h-cobordism if ∂0W → W and ∂1W → Ware weak equivalences. Last time we showed that if w ≥ 6 then W admits ahandle decomposition without 0, 1, w − 1, w-handles.

Lemma 17.1. Let 2 ≤ q ≤ w − 3, and let W be an h-cobordism of dimensionw ≥ 6. Then W admits a handle decomposition

W = ∂0(W )× I +∑Iq

(φqiq ) +∑Iq+1

(φq+1iq+1

).

We’ll give a proof in the case when W is simply-connected. The two maintechniques will be handle sliding and the Whitney trick.

Proof. Without loss of generality we may assume an handle decomposition with-out 0, 1, w − 1, w-handles. It suffices to how to trade the lowest index i-handle(φii) for a i+ 2-handle.

We start by picking a trivial embedding ψi+1 : ∂Di+1 ×Dw−i−1 → ∂i(Wi).

This automatically satisfies (b) of the handle exchange lemma, and we willmodify it to satisfy (a). That is, the attaching sphere ψi+1(∂Di+1 × 0) isisotopic in ∂1(W i) to one intersecting transverse sphere of φii once transversally.

Since W is a h-cobordism, Hi(W,∂0(W )) = 0 and there are no i′-handles fori′ < i. This shows that we can write

[φii] =∑j∈Ii+1

ajd[φi+1j ]

as a Z-linear combination of attaching spheres of (i+ 1)-handles.We will “add” this to ψi+1, and it suffices to show how to add ±[φi+1

j ] to

[ψi+1(∂Di+1×0)]. Let Si+1

j be the parallel translate of φi+1i (∂Di+1×0) in

∂1(W i). Pick an embedded arc from a point in ψi+1(∂Di+1 × 0) to a point

in Si+1

j and thicken it to 1×Di, with

0 ×Di ⊆ ψi+1(∂Di+1 × 0), 1 ×Di ⊆ Si+1

j .

use this to replace ψi+1(∂Di+1 × 0) with the embedded connected sum of

ψi+1(∂Di+1×0) and Si+1

j , lying in ∂1(W i). Since Si+1

j bounds a (i+ 1)-disk

in ∂1(W i), we see that

ψi+1(∂Di+1 × 0)#Si+1

j

Math 272x Notes 51

is isotopic to ψi+1(∂Di+1×0), is still trivial in ∂1(W i+1), and can be extededto an embedding ψi+1 of ∂Di+1 ×Dw−i−1.

Using this, we can find a ψi+1 that is trivial in ∂1(W i+1), and has intersectionnumber 1 with the transverse sphere of (φii). Now this can have 15 intersectionswith 8 positive and 7 negative signs, but the Whitney trick says that we canisotope ψi+1 further such that hte number of intersection points is 1. So we cannow handle exchange to replace (φii) with (ψi+2).

17.2 Wh1(π1)

Now we have

W = (∂0W × I) +∑Iq

(φqj) +∑Iq+1

(φq+1j ).

Since W is an h-cobordism, we have H∗(W,∂0(W )) = 0. This shows that

∂q : Z[Iq+1]→ Z[Iq]

is an isomorphism.From this we get an invertible matrix D with integer entries, well-defined

up to

(i) multiplying rows or columns by −1,

(ii) interchanging rows or columns,

(iii) handle addition or cancellation, i.e., adding or subtracting a row and acolumn like

D ←→(D 00 1

),

(iv) handle slides, adding a multiple of a row to another row (or column).

If π1(W ) 6= 0, then passing to the universal cover gives a matrix D withentries in the group ring Z[π1].

Note that if n ≥ 3, then elementary matrices are commutators,

eik(ab) = [eij(a), ejk(b)].

Moreover,

colimn→∞

En(R) ⊆ (commutator subgroup of colimn→∞

GLn(R)).

This is because([g, h]

idn

)=

(g

g−1

)(h

h−1

)((hg)−1

hg

),(

gg−1

)=

(id g

id

)(id−g−1 id

)(id g

id

)(− id

id

).

Math 272x Notes 52

So we can take care of (ii), (iii), (iv) simultaneously, if we consider D aslying in

K1(R) = colimn→∞

GLn(R)ab = colimn→∞

H1(GLn(R)).

To take care of (1), we define

Definition 17.2. Wh1(G) = K1(Z[G])/(±g).

Definition 17.3. For a cobordism W , the Whitehead torsion is defined as

τ(W ) = [D] ∈Wh1(π1(W )).

The conclusion is the inverse direction of the following theorem.

Theorem 17.4. If W is an h-cobordism of dimension w ≥ 6, then W ∼=∂0(W )× I relative to ∂0(W ) if and only if τ(W ) = 0.

17.3 Poincare conjecture

Lemma 17.5. Wh1(e) = 0.

Proof. Because

GL1(Z) → GLn(Z)det−−→ GL1(Z)

is the identity, it suffices to show that SLn(Z) = En(Z) as n→∞. This is doneby the Euclidean algorithm.

Corollary 17.6. If M is an m-dimensional closed manifold with M ' Sm,then M is homeomorphic to Sm.

Proof. Take a DmqDm embedded in M . The complement W is an h-cobordismand simply connected. This shows that W ∼= Sm−1 × I. Then

M ∼= Dm0 ∪id (Sm−1 × I) ∪f Dm

1∼= Dm

0 ∪f Dm1 .

But every homeomorphism of Sm−1 extends over Dm by the Alexander trick.

What we have actually proved is that there is a map

π0(Diff∂(Dm−1)) ∼= π0(Diff+(Sm−1))

Θm =

set of m-dimensional closed

manifolds homotopy equivalent to Sm

,

since Diff+(Sm−1) ' SO(m)×Diff∂(Dm−1).

Math 272x Notes 53

18 October 13, 2017

Last time we learned that

π0(Diff∂(Dn)) Θn+1,

where Θn is the set of exotic spheres.

18.1 A Brieskorn sphere

Theorem 18.1 (Brieskorn). For m > 1, the manifold

Σ = z ∈ C2m+1 : z30 + z5

1 + z22 + · · ·+ a2

2m = 0, |z| = 1

is homeomorphic, but not diffeomorphic, to S4m−1.

Recall that the signature σ(m) of a orientable 4m-manifold M is the num-ber of positive eigenvalues minus the number of negative eigenvalues of theintersection form of H2m(M ;R).

Theorem 18.2 (Kervaire–Milnor). Let Σ be homeomorphic to S4m−1 and sup-pose Σ = ∂W with W parallelizable. Then Σ is diffeomorphic to S2m−1 if andonly if

1

8σ(W ) ≡ 0 (mod

3 + (−1)m+1

222m−2(22m−1 − 1) Num

(B2m

4m

)).

So we are going to show that Σ is a topological sphere and σ(W ) = ±8. Thenotation we are going to use is

f(z) = za00 + · · ·+ zann n ≥ 2,

V (f) = z ∈ Cn+1 : f(z) = 0,

and Σ = V (f) ∩ S2n+1. Also, Caj is a cyclic group of order aj with fixedgenerator ω.

Because Σ is inside a sphere, Alexander duality gives a good way to under-stand it by understanding the complement. There is a map ϕ : z 7→ f(z)/|f(z)|.

S2n+1 \ Σ Cn+1 \ V (f)

S1 S1

ϕ ϕ

Theorem 18.3 (Milnor). These maps are smooth fiber bundles, and the topmap is a fiberwise homotopy equivalence. Moreover, F 0 = ϕ−1(1) is a smoothmanifold with boundary Σ, and both F0 and Σ are connected and simply con-nected.

The group G = Ca0 × · · · ×Can acts on F0 by multiplying e2πi/aj in the jthcoordinate. Then π1(S1, 1) = T acts via T → G; t 7→ w = w0 × · · · × wn.

Math 272x Notes 54

Proposition 18.4. There is a homotopy equivalence

F0 '∨µ(f)

Sn,

where µ(f) =∏j(aj − 1).

Recall thatA ∗B = C(A)×B qA×B A× C(B).

For instance, (∆0)∗(n+1) = ∆n.

Proof. Let J = Ca0 ∗ · · · ∗ Can . Then there is a map

J → ϕ−1(1); (tjwtjj ) 7→ (e2πirj/aj t

1/rjj )

that is a G-equivariant deformation retract. Then F0 is (n−1)-connected. Also,we can compute

H∗(F0) ∼= H∗(J) = H∗(Z[G]→ Z[G]∂jσ/(∂jσ − ωj∂jσ)→ · · · ).

Then Hi(F0) = 0 for i > n, and

Hn(F0) ∼= Z[G]η ∼=⊗j

Z[Cnj ]/(1 + wj + · · ·+ ωaj−1j ),

where η =∏j(1− ωj).

Corollary 18.5. F0 is parallelizable.

Proof. Because dimF0 = 2n and H2n(F0) = 0, F0 has no compact component.But F0 is stably parallelizable because it is embedded in Cn+1 with trivialnormal bundle.

Corollary 18.6. Hi(S2n+1 \ Σ) ∼=

coker(1− ω) i = n

ker(1− ω) i = n+ 1

0 otherwise.

Proof. Use the Serre spectral sequence. Here, the coefficients aren’t trivial, butwe know exactly how π1 acts.

Let ∆(x) = det(x− ω) be the characteristic polynomial.

Corollary 18.7. Σ is a topological sphere if and only if |∆(1)| = 1.

Proof. We know that |∆(1)| = 1 if and only if S2n+1 \ Σ has the homology ofa point. Now use Alexander duality and Poincare duality to conclude that Σis a homology sphere. Because Σ is simply connected and we have generalizedPoincare conjecture, we get the result.

Math 272x Notes 55

Now we use the splitting of Hn(F0) to write

∆(x) =∏

0<rj<aj

(x−

∏j

e2πirj/aj).

Example 18.8. If n = 2m and f(z) = z30 + z5

1 + z22 + · · ·+ z2

2m, then

∆(x) = Φ15(x) = x8 − x7 + x5 − x4 + x3 − x+ 1.

So Σ is a topological sphere in this case.

18.2 Signature of the sphere

Theorem 18.9 (Pham). The intersection form on Hn(F0) is

〈g, h〉 = ε(h−1gη)

where ε : Z[G]→ Z is specified as ε(1) = −ε(ω).

Given a linear map λ : C[G]→ C, define

λ =∑g∈G

λ(g)g ∈ C[G].

Corollary 18.10. σ(F0) = σ+(F0)− σ−(F0), where

σ+(F0) = #(χ : G→ C×) : χ(εη) > 0,σ−(F0) = #(χ : G→ C×) : χ(εη) < 0.

Proof. We know that χ is a basis for C[G], and we always have

χη = χ(η)χ

where g = g−1. SoHn(F0)⊗ C ∼= C〈x : x(η) 6= 0〉.

By orthogonality, we have

〈χη, χ′η〉 =

|G|χ(εη) χ = χ′

0 otherwise.

So we can just count these eigenvalues.

Corollary 18.11. If n is even, then σ(F0) = N+ −N− where

N+ = #(r0, . . . , rn : 0 < rj < aj , 0 <∑jrjaj

mod 2 < 1,

N− = #(r0, . . . , rn : 0 < rj < aj , 1 <∑jrjaj

mod 2 < 2.

Math 272x Notes 56

Proof. We check εη = (−1)n(n+1)/2(η + η). So you can check

χ(εη) = (−1)n/2<(χ(η)) = · · · = |χ(η)| sin(π∑j

rjaj

).

Then you can translate to the statement.

Proof of the theorem. We have n = 2m, a0 = 3, a1 = 5, aj = 2 for j > 1. So

r0

3+r1

5+

2m− 1

2=

10r0 + 6r1 + 15(2m− 1)

30=

10r0 + 6r1

30+

(−1)m+1

2(mod 2).

Here, 10r0 + 6r1 ∈ 16, 26, 22, 32, 28, 38, 34, 44 because r0 ∈ 1, 2 and r1 ∈1, 2, 3, 4. This shows that σ(F0) = 8(−1)m.

Math 272x Notes 57

19 October 16, 2017

In the last lecture I gave, we showed that

π0(Diff∂(Dm−1)) Θn+1

for n ≥ 5. Today I am going to give the Whitney trick, and the Casson trick indimension 4.

19.1 Whitney trick

Let N be a smooth manifold and M,X ⊆ N , with m + x = n. Also assumeM t X. Pick p0, p1 ∈M ∩X which are in the same path components of M andof X.

Our goal is to remove p0 and p1 by isotopic X. Pick embedded paths γMon M and γX on X connecting p0 and p1, and avoiding all other intersectionpoints. Pick a Riemannian metric g in N such that

(i) M and X are totally geodesic near ϕM and ϕX ,

(ii) M ⊥ X at p0 and p1.

Pick basis ζ1(0), . . . , ζx(0) of Tp0(X) = Tp0(M)⊥ with ζx(0) = dγx/dt. Like-wise pick a basis ξ1(0), . . . , ξm(0) or Tp0(M) = Tp0(X)⊥ with ξm(0) = dγm/dt.They collectively form a basis of Tp0(N). Parallel transport the basis along γXand along γM to get two bases

ζX1 (1), . . . , ζXx (1), ξX1 (1), . . . , ξXm(1) and ζM1 (1), . . . , ζMx (1), ξM1 , . . . , ξMm (1)

of Tp1(N). Assume that they are oppositely oriented. Let B2 be the “bigon”,so that we have an embedding

f : ∂B2 → N ; f |∂+(B2) = ϕX , f |∂−(B2) = ϕM .

Lemma 19.1. If m,x < n − 2 and π1(N) = 0, then π1(N \ (M ∪X)) = 0. Ifmoreover n ≥ 5, then f may be extended to an embedding ϕ : B2 → N such thatϕ(int(B2)) ⊆ N \ (M ∪X).

Proof. If codimension of M and X are at least 3, then every loop or null-homotopy generically avoids M and X. Now f is null-homotopic in N \(M∪X),and generically null-homotopy is embedded if n ≥ 5.

We want to thicken B2 by flowing along normal vector fields. The tangentspace TB2 is span(ζMx (t), ξMm (t)) over ϕM , and span(ζXx (t), ξMm (t)) over ϕX .

Lemma 19.2. We can extend (ξM1 (t), . . . , ξMm−1(t)) and (ζX1 (t), . . . , ξXx−1(t)) toa trivialization of the normal to B2, such that ξM over γX are orthogonal to Xand ζX over γM are orthogonal to M .

Math 272x Notes 58

Proof. Basically we are going to extend ζX over γX to over γM , and then fill inB2. Then we can do the same for ξM .

Pick a vector field ρ on B2, which is orthogonal to ∂−(B2) (correspondingto γM ). This is possible because the two frames in (TM)⊥

(ζX1 (0), . . . , ζXx−1(0), ρ0 = dγXdt ) and (ζX1 (1), . . . , ζXx−1(1), ρ1 = −dγxdt )

have the same orientation.So we get an extension over ϕm of ξX1 , . . . , ξ

Xx−1. We want to extend this to

an (x− 1)-frame in (TB2)⊥. The obstruction to extending is

π1(O(n− 2)/O(m− 1)) = 0

because m− 1 ≥ 2. So we can indeed ζX .Now there is no further to extending ξM to trivializations of (span(ζX))⊥ ⊆

(TB2)⊥. Then ξM over γX are perpendicular over γX .

Denote these extensions by (ζX1 , . . . , ζXx−1, ξ

M1 , . . . , ξMm−1) over B2. We can

extend them a bit outside B2 and exponentiate to get standard coordinatesaround B2. This gives a local model

M = D− × Rm−1 × 0, X = D+ × 0 × Rx−1

in Rn = R2 × Rm−1 × Rx−1.

Theorem 19.3 (Whitney trick). Let M and X be submanifolds of codimension≥ 3 in a simply-connected n = (m + x)-dimensional manifold N . AssumeM t X and p0, p1 ∈ M ∩ N are points in the same path component of Xand M . If they have opposite signs, then there there is a compactly supportedambient isotopy ϕt such that M ∩ ψ1(X) = M ∩N \ p0, p1.

19.2 Casson trick

In dimension 4, everything in this argument goes wrong.

(i) The complement of M ∪X in N might not be simply-connected.

(ii) A generic 2-disk in a 4-manifold is not embedded.

(iii) There might be framing obstructions, lying in π1(O(2)/O(1)) = Z.

It turns out that you can solve (i) and (iii) at the expense of (ii). Theproblem (iii) can be solved using some boundary twisting. Solving (i) is calledthe Casson trick.

There is something called the finger move. Suppose Σ ⊆ N is a surfaceand there is another surface. Let γ be an arc connecting p ∈ Σ and the othersurface. Then we can extend Σ along the path γ and get another surface Fγ(Σ).This is going to have two more intersection points.

Suppose p1 is near p0 on Σ. Then there is an element ζ that rotates aroundΣ in p0, and an element η that follows this path γ.

Math 272x Notes 59

Lemma 19.4. π1(N \ Fγ(Σ)) = π(N \ Σ)/([z, zη]).

Theorem 19.5 (Casson). Let N be simply connected, and (D2, ∂D2)# (N, ∂N)embedded on ∂. If there exists a β ∈ H2(N) such that [D2] ·β = 1, then D2 maybe modified by finger moves such that the complement is simply-connected.

Proof. π1(N \ D2) is normally generated by meridians. Then [z] generatesH1(N \ D2). This means that π1(N \ D2) is generated by commutators. Soit is normally genreated by [z, zw], which can be killed by finger moves.

Now we can find an immersed disk with intersections. We can then findsmaller disks that come from the intersections, and so they can be modified toget smaller disks. So you get the whole system of disks branching out to smallerdisks, which tries to look like a Whitney disk. In the topological category, thiswhole thing actually contains a topological disk.

Math 272x Notes 60

20 October 18, 2017

In the next 5 lectures, we are going to show that the map Diff∂(Dn) Θn+1 issurjective if n ≥ 5 and injective if n ≥ 7. Also, we are going to compute rationalhomotopy π∗(Diff∂(Dn))⊗Q.

20.1 Motivation for algebraic K-theory

You can think of algebraic K-theory as

• the natural home for algebraic topology invariants,

• the algebro-geometric version of topological K-theory, or

• homotopy-coherent way of doing group completioins or splitting short ex-act sequences.

Definition 20.1. Let R be a commutative ring. A vector bundle over SpecRis a sheaf locally isomorphic to O⊕nSpecR.

Then there is a correspondence between vector bundles over SpecR andfinitely generated projective R-module. This is given by E 7→ Γ(E) and M 7→M(Df ) = Mf . We can then group complete this. Here, we don’t need to splitshort exact sequences, because they are already split.

Definition 20.2. K0(R) is the group completion of projective finitely generatedmodules with ⊕.

People realized that this fits with the K1 we defined last time. For example,the following is exact.

K1(Fp)→ K1(Z(p))→ K1(Q)→ K0(Fp)→ K0(Z(p))→ K0(Q).

Then people found out spaces whose homotopy groups are the K-groups. Thatis, there is a space K(R) such that Ki(R) = πiK(R).

20.2 Group completion

Definition 20.3. A Γ-space is a functor Γ → Top, where Γ is the categoryof pointed finite sets, such that the map X(n) →

∏nX(1) induced by n =

∗, 1, . . . , n → ∗, i = 1 is a weak equivalence. (This implies that X(0) = ∗.)

Here, the map

X(1)×X(1) X(2) X(1)'

is sort of like multiplication, and

∗ X(0) X(1)'

is the unit.

Math 272x Notes 61

Example 20.4. If M is a commutative unital monoid, then XM : n 7→ Mn isa Γ-space.

Example 20.5. Let R be a ring, and consider XR that maps n to the classifyingspace of (the category with objects n-tuples of finitely generated projective leftR-modules, and morphisms n-tuples of R-linear isomorphisms).

Now there is a functor ∆→ Γop given by

[n] = 0→ 1→ · · · → n 7→ 0→ 1, 1→ 2, . . . , n− 1→ n, ∗ ∼= n,

and the morphisms pulling back by looking at the gaps. We can use this to geta simplicial object

B•X : ∆op → ΓX−→ Top.

This is actually something like a bar resolution.

Example 20.6. Let us look at B•XM . If you look at a face map [n]→ [n+ 1],the corresponding map will be just multiplying the two copies of M that theface map skips.

Theorem 20.7 (Segal). The geometric realization BX = |B•X| is an infiniteloop space.

Here is how you do this. There is a X(1) × ∆1 → BX that is a point onthe boundary of ∆1. So we get X(1) → ΩBX. Iterate this by noting thatS 7→ (T 7→ X(S ∧ T )) is a Γ-space of Γ-spaces. This is B(1)X : S 7→ BXS =(T 7→ X(S ∧ T )). Then we get BX → ΩBB(1)X, and then go on.

Definition 20.8. We define KΩB(R) = ΩBXR, which is actually an infiniteloop space. Then we define Ki(R) = πiK

ΩB(R).

Example 20.9. Let us compute K0(R) = π0ΩBXR = π1BXR. But this BXR

has a cell-decomposition and we know exactly what they are. The 1-cells areobjects in the category of finitely generated finitely generated projective leftR-modules. Then 2-cells are isomorphisms in this category. So π1 is generatedby γP , and the relation is that γP ' γP ′ if P ∼= P ′.

Let MR be given by the classifying space of objects n-tuples of R-modulesand morphisms R-linear isomorphisms. This has a natural commutative andunital structure coming from ⊕.

Theorem 20.10 (Segal–McDuff). If M is homotopy commutative associativeunital topological monoid, then

H∗(ΩBM) ∼= H∗(M)[π−10 ].

Applying this to MR, which has BMR ' BXR gives

H∗(ΩBXR) ∼= H∗(MR)[π−10 ] ∼= H∗

(∐[P ]

BGL(P )

)[π−1

0 ]

∼= H∗

(∐n≥0

BGLn(R))

[π−10 ] ∼= lim−→H∗

(∐n≥0

BGLn(R)

).

Math 272x Notes 62

So we recover

K1(R) = π1(KΩB(R)) = H1(KΩB(R)) ∼= lim−→n

H1(BGLn(R)).

Math 272x Notes 63

21 October 20, 2017

Today I want to state the theorems of Igusa and Waldhausen. Last time Iexplained how to construct the spaces K(R). Today we will spend some timeconstructing K(SΩX).

21.1 S•-construction

The input for this is a Waldhausen category.

Definition 21.1. A Waldhausen category C is a pointed category C withtwo classes of morphisms, cofibrations and weak equivalences, satisfying

(i) isomorphisms are cofibrations and weak equivalences,

(ii) ∗ → A are cofibrations,

(iii) cofibrations and weak equivalneces are closed under composition,

(iv) if B → A is a cofibration and B → C is any map, the pushout C ∪B Aexists and C → C ∪B A is a cofibration,

(v) in the above situation, if A→ A′, B → B′, C → C ′ are weak equivalenceswith B′ → A′ cofibration, then C∪BA→ C ′∪B′A′ is a weak equivalence.

Let Ar(n) be the category of arrows in

0→ 1→ · · · → n

so that there are n(n+ 1)/2 objects.

Definition 21.2. Let SnC be the subcategory of Fun(Ar(n), C) such that

• F (i→ i) = ∗,• F (i→ j) → F (i→ k) is a cofibration,

• the diagram

F (i→ j) F (i→ k)

F (j → j) = ∗ F (j → k)

is a pushout square.

Example 21.3. Let us write out what S2C is. They have objects given bydiagrams

∗ F (0, 1) F (0→ 2)

∗ F (1→ 2)

∗.

Math 272x Notes 64

Here, you can think of ∗ → F (0 → 1) → F (0 → 2) is a filtered object, withF (0→ 1) and F (1→ 2) the associated graded.

Here [n] 7→ Ar(n) is a cosimplicial set, so [n] 7→ SnC is a simplicial category.Consider wSnC ⊆ SnC the subcategory with the same objects but morphismsnatural weak equivalences. Then [n] 7→ wSnC is also a simplicial category.

Definition 21.4. Ω|[n] 7→ |N•wSnC|| = KS•C.

For example, take P (R) the finitely generated projective left R-modules,with cofibrations inclusions with projective cokernels and weak equivalencesisomorphisms.

Theorem 21.5 (Quillen–Waldhausen). KΩB(R) ' KS•(P (R)).

As another example, take FinSet∗ be the pointed finite sets with cofibrationinclusions and weak equivalences isomorphisms. In this case, it is easy to seethat

wSn(FinSet∗) ' FinSetn∗

because the diagram is just determined by F (i→ i+1). So we can trace throughthe definitions and get

KS•(FinSet∗) ' ΩB

(∐n≥0

BΣn

).

Theorem 21.6 (BPQS). ΩB(∐n≥0BΣn) ' Ω∞S.

21.2 Algebraic K-theory of spaces

Let X be a space.

Definition 21.7. A retractive finite space over X is a triple (Y, i, r) suchthat i : X → Y is a cofibration, r : Y → X is a retraction so that r i = idX ,and (Y,X) is homotopy equivalent to a relative finite CW-complex.

Let Rf (X) be the Waldhausen category with objects retractive finite spacesover X. The cofibrations will be cofibrations and weak equivalences will be weakequivalences.

Definition 21.8. The algebraic K-theory of spaces of X is

A(X) = KS•(Rf (X)).

Example 21.9. Let us compute π0A(∗). Note that π0 is given by [ pointedspaces, weakly equivalent to a finite CW-complex with a retraction to ∗. Thereare relations [Y ] = [Y ′] if Y ' Y ′ and [Y ′] = [Y ] + [Y ′′] if there is a cofibersequence Y → Y ′ Y ′′.

But you can do this. We have

0 = [∗] = [Dk] = [Sk−1] + [Sk]

Math 272x Notes 65

because Sk−1 → Dk → Sk. Then we see that [Sk] = (−1)k[S0]. Any finitepointed complex Y splits as

[Y ] =∑k≥0

#(k-cells)[Sk] = χ(Y, ∗).

So π0A(∗) ∼= Z.

There is also a group completion construction of A(X) for path-connectedX. Let GX be the Kan loop group, or the topological group weakly equivalentto ΩX.

Definition 21.10. GLn(Σ∞+ GX) is the homotopy colimit as k → ∞ somesubspace of

∏n Ωk(

∨n S

k ∧GX+). If k ≥ 2, then

π0Ωk(∨n

Sk ∧GX+

)∼=⊕n

Z[π1(X)]n = Mn(Z[π1]),

so we take the subspace corresponding to GLn(Z[π1]).

Theorem 21.11 (Waldhausen). If X is path-connected, then

A(X) ' ΩB

(∐n≥0

BGLn(Σ∞+ GX)

).

The advantage of this is that there is a map

Σn oGX → GLn(Σ∞+ GX)

and induces the map

ΩB

(∐n≥0

B(Σn oGX)

)→ ωB

(∐n≥0

BGLn(Σ∞+ GX)

)' A(X).

The first thing is Ω∞Σ∞X+ = QX+, so we have a map QX+ → A(X). Thismap actually comes from a map of spectra Σ∞X+ → A(X).

Definition 21.12. We define WhDiff(X) = cofib(Σ∞X+ → A(X)), and wedefine

WhDiff(X) = Ω∞WhDiff(X).

Theorem 21.13 (Waldhausen). A(X) ' QX+ ×WhDiff(X).

21.3 Moduli spaces of h-cobordisms

The s-cobordism classified h-cobordisms starting at M with dimM ≥ 5. Here,we figured out that

h-cobordismsstarting at M

/diff. rel M ∼= Wh1(π1(M)).

Now we can try to figure out what the automorphisms are. At least, theautomorphisms of the trivial h-cobordism M →M × I are Diff(M × I relM ×0). This generalizes to ∂M 6= ∅.

Math 272x Notes 66

Definition 21.14. We define C(M) = Diff(M × I relM × 0 ∪ ∂M × I).

Then we can define the moduli space of h-cobordisms as

H(M) = Wh1(π1(M))×BC(M).

There is a map H(M) → H(M × I) induced by homomorphisms C(M) →C(M × I).

Definition 21.15. We define the stable moduli space as

H (M) = hocolimh→∞

H(M × Ik).

Theorem 21.16 (Igusa). The map H(M)→H (M) is min(m−13 , m−5

2 )-connected.

Theorem 21.17 (Waldhausen). H (M) ' Ω WhDiff(M).

Math 272x Notes 67

22 October 23, 2017

Last time we had this claim that there is a map

H(M)→ Ω WhDiff(M)

which is approximately m3 -connected. Today we will use this to prove that

π0(Diff∂(Dn))→ Θn+1 is and isomorphism if n ≥ 7.

22.1 Computation of π0H(X)

Recall that for X path-connected,

QX ' ΩB(∐

n≥0B(Σn oGX))

ΩB(∐

n≥0BGLn(Σ∞+ GX))' A(x)

Q(Bπ1)+ ' ΩB(∐

n≥0B(Σn o π1))

ΩB(∐

P BGL([P ]))' K(Z[π1]).

Then we can take homotopy fibers of the vertical maps, apply homotopy groupsand take cokernels of the horizontal maps.

......

...

πSn+1(Bπ1/X) πnF Fn(X) 0

0 πSn (X+) πnA(x) πn(WhDiff(X)) 0

πSn (Bπ1+) Kn(Z[π1]) Wn(π1) 0

......

...

Here we don’t know if the right vertical sequence is exact.

Remember that π0A(∗) ∼= Z, and this generalizes to π0QX+

∼=−→ π0A(X). Sothe first possible non-zero πn WhDiff(X) is π1.

Lemma 22.1. QX+ → Q(Bπ1)+ and A(X)0 → K(Z[π1])) are 2-connected.

Understandand includeproof

Understandand includeproof

So we can include this to get the following diagram:

0 πS1 (X+) π1A(X) π1 WhDiff(X) 0

0 πS1 (Bπ1)+ K1(Z[π1]) W1(π1) 0

∼= ∼= ∼=(∗)

Math 272x Notes 68

So to compute π1 WhDiff(X), we only need to figure out the cokernel of (∗) :πS1 (Bπ1)+ → K1(Z[π1]).

We can compute πS1 (Bπ1)+ using the Atiyah–Hirzebruch spectral sequence.Then E2

p,q = Hp(Bπ1;πSq ) converges to this, and so we get

πS1 (Bπ1)+∼= H1(Bπ1)⊕ Z/2Z ∼= (π1)ab ⊕ Z/2Z.

Now the map (∗) is just the inclusion of [g] and [−1], so it generates the samesubgroup of K1(Z[π1]) as (±g). This shows that

π0H(X) ∼= π1 WhDiff(X) ∼= W1(π1) ∼= Wh1(π1).

This is just the s-cobordism theorem again.

22.2 Computation of π1H(X)—the HWI sequence

Let us now compute π2 WhDiff(X).

K3(Z[π1]) W3(π1) 0

πS3 (Bπ1/X) π2F F2(X) 0

0 πS2 (X+) π2A(X) π2 WhDiff(X) 0

πS2 (Bπ1)+ K2(Z[π1]) W2(π1) 0

0 0 0

Lemma 22.2. π2F = H0(Bπ1; (π2 ⊕ Z/2Z)[π1])

Proof. We have, by the group completion theorem,

colimn→∞BGLn(Σ+GX) A(X)

colimn→∞BGLn(Z[π1]) KZ[π1].

Vertical maps have the same relative homology in positive degress, so

π2F ∼= π3(K(Z[π1])) ∼= H3(K(Z[π1]), A(X))∼= colim

n→∞H3(BGLn(Z[π1]), BGLn(Σ+GX)).

Now there is a relative Serre spectral sequence coming fromBMn(Q0GX+)→BGLn(Σ∞+ )→ BGLn(Z[π1]) mapping to ∗ → BGLn(Z[π1])→ BGLn(Z[π1]).

Math 272x Notes 69

This spectral sequence is going to be

E2p,q = Hp(BGLn(Z[π1]), Hq(∗, BMn(Q0GX+))),

but the coefficient is Hq−1(BMnQ0GX+). This has first non-zero group at q = 3by the Hurewicz theorem, and it is

H2(BMnQ0GX+) ∼= π2(BMnQ0GX+) ∼= MnQ0GX1) = Mn(π1(Q0GX+)).

Here, π1Q0GX+∼=⊕

γ∈π1πS2 (X〈1〉+) which are π2(X)⊕ Z/2Z. So we get

π2F = H0(BGLn(Z[π1]);Mn((π2 ⊕ Z/2Z)[π1])),

and the trace map to H0(Bπ1; (π2 ⊕ Z/2Z)[π1]) is an isomorphism.

Lemma 22.3. πS3 (Bπ1/X) ∼= H0(Bπ1; (π2)[1]).

Proof. We have πS3 (Bπ1, X) ∼= H3(Bπ1, X) by Atiyah–Hirzebruch. We alsohaveH3(Bπ1, X) ∼= π3(Bπ1, X)/π1 by Hurewicz. This is isomorphic to π2(X)/π1

∼=H0(Bπ1;π2).

Once you further prove that some maps are injective, you get that the lastvertical sequence is exact.

Theorem 22.4 (Hatcher–Wagoner–Igusa). The sequence

K3(Z[π1])→ H0(Bπ1; (π2 ⊕ Z/2Z))

H0(Bπ1; (π2 ⊕ Z/2Z)[1])→ π2 WhDiff(X)→ K2(Z[π1])

πS2 (Bπ1)+→ 0

is exact.

Corollary 22.5. If π1(X) = e with dimX ≥ 7, then π0C(X) = π1H(M) =π2 WhDiff(X) = 0.

Proof. The left term vanishes automatically. The right term also vanishes be-cause

πS2∼= Z/2Z→ K2(Z) ∼= Z/2Z→ K2(Rtop) ∼= π2ko ∼= Z/2Z

is an isomorphism.

Math 272x Notes 70

23 October 25, 2017

Today is the last lecture on algebraic K-theory.

23.1 Pseudo-isotopy implies isotopy

Recall we have the Hatcher–Wagoner–Igusa sequence. For M of dimension ≥ 7,

K3(Z[π1])→ H0(π1; (π2 ⊕ Z/2Z)[π1])

H0(π1; (π2 ⊕ Z/2Z)[1])→ π1 WhDiff(M) ∼= π0C(M)→ K2(Z[π1])

πS2 (Bπ1)+→ 0

is exact. If π1(M) = e, then we have established that π0C(M) = 0. I want tostate this more concretely.

Definition 23.1. For two diffeomorphisms f, g : M → M , f is pseudo-isotopic to g if thee is a diffeomorphism F : M × I → M × I such thatF |M×0 = f and F |M×1 = g.

This is not an isotopy because we don’t have the condition that F preservesthe t-coordinate.

Corollary 23.2. If m ≥ 7 and π1(M) = e, then f being pseudo-isotopic tog implies f isotopic to g.

Proof. Take an arbitrary pseudo-isotopy F : M × I → M × I. Then F ′ =(f × id)−1 F is a pseudo-isotopy from id to f−1 g. Then F ′ is a concordancediffeomorphism, i.e., in Diff(M × I,M × 0). Because π0(C(M)) = 0, there isan isotopy F ′t from F ′ to idM×I . But then F ′t |M×1 is an isotopy of diffeomor-phisms of M from id to f−1 g. Composing with f gives an isotopy from f tog.

23.2 π0(Diff∂(Dn)) and Θn+1

Recall that Θn+1 is the space of oriented homotopy (n+ 1)-spheres up to orien-tation preserving diffeomorphisms. There is an abelian monoid structure, whichis to take two spheres and taking the connected sum. Here are some things wehave proven.

• this connected sum is well-defined up to orientation-preserving diffeomor-phism.

• it is commutative and associative.

• there is a unit [Sn+1].

Lemma 23.3. If n ≥ 4, this is an abelian group.

Proof. Let Σ be any oriented homotopy (n+1)-sphere. We take Σ×I and removea neighborhood of ∗ × I. Let this be W . Then W is an (n + 2)-dimensionalmanifold with

∂W = Σ#Σ,

Math 272x Notes 71

where Σ is the Σ with opposite orientation. Now take w ∈ W and remove alittle dist. Then W is a h-cobordism from Σ#Σ to Sn+1, so Σ#Σ ∼= Sn+1.

Given f ∈ Diff+(Sn), we can construct a sphere Sn+1f = Dn+1 ∪f Dn+1.

This gives a map

π0(Diff∂(Dn)) π0(Diff+(Sn)) Θn.∼=

Here, the first isomorphism follows from Diff+(Sn) ' SO(n + 1) × Diff∂(Dn),and the second surjection follows from the h-cobordism theorem.

Lemma 23.4. π0(Diff∂(Dn)) Θn+1 is a group homomorphism.

Proof. Taking the connected sum along the glued parts. Then we get

Sn+1f #Sgn+ 1 ∼= Sn+1

f\g ,

where f\g is the juxtaposition, putting them next to each other. But note thatf g is isotopic to (f\1) (g\1) = f\g.

Now we want to show that π0(Diff∂(Dn)) Θn+1 is an isomorphism. Itsuffices to show that if Sn+1

f∼= Sn+1 then f is isotopic to id.

Proposition 23.5. If n ≥ 6 and f ∈ Diff+(Sn) and Sn+1f∼= Sn+1, then f is

isotopic to id.

Proof. We have i1 : Dn+1qDn+1 → Sn+1f and i2 : Dn+1qDn+1 → Sn+1, where

the first disk is mapped to a little disk around the origin in one hemisphere andthe second disk is mapped to the other hemisphere.

The embeddings i2 and g i1 are orientation-preserving. By isotopy exten-sion, we can assume that the following diagram commutes.

Dn+1 qDn+1 Dn+1 qDn+1

Sn+1f Sn+1

i1 i2

g

The restricting the complement gives a g : Sn×I → Sn×I such that g|Sn×0 =id and g|Sn×1 = f−1. This shows that f−1 is pseudo-isotopic to id, and so fis isotopic to id.

Corollary 23.6. If n ≥ 7, then π0(Diff∂(Dn)) ∼= Θn+1.

Cerf proved that you can actually improve this to n ≥ 5.

Theorem 23.7 (Kervaire–Milnor). If n ≥ 2, then Θn+1 is finite.

The strategy is to prove that each Σ ∈ Θn+1 admits a stable framing. Onceyou’ve done that, send it to p(Σ), the collection of bordism classes of Σ with achoice of stable framing. This is a subgroup of πSn+1, and get

0→ bPn+2 → Θn+1 → πSn+1/p(Sn+1)→ 0.

Here bPn+1 is those Σ which admits a stable framing such that it bounds aframed manifold. Then we can just prove that bPn+2 is finite, but it is hard.

Math 272x Notes 72

24 October 27, 2017

Last time we proved that π0(Diff∂(Dn)) ∼= Θn+1 is finite if n ≥ 5. Today we’lluse this to compute π∗(Diff∂(Dn))⊗Q.

24.1 Concordances rationally

Recall that that C(M) = Diff(M × I relM × 0 ∪ ∂M × I). If we restrict toM × 1 we get a fiber sequence

Diff∂(Dn+1)→ C(Dn)→ Diff∂(Dn).

So let us compute C(Dn) rationally first.The Igusa–Waldhauset theorems give

π∗C(Dn)⊗Q ∼= π∗+2 WhDiff(∗)⊗Q

for ∗ ≤ min(n−73 , n−9

2 ). This is the concordance stable range. Now there is a

waldhausen spitting A(∗) ' S×WhDiff(∗).Now note that π∗S ⊗ Q is Q if ∗ = 0 and 0 otherwise, and S → A(∗) is an

π0-isomorphism. So we get

π∗WhDiff(∗) =

0 ∗ = 0

π∗A(∗) otherwise.

Now ∏n2

Q0S0 ' Q∗ → GLn(S)→ GLn(Z)

and so A(∗)→ K(Z) is a Q-isomorphism. The summary is that

πiC(Dn)⊗Q ∼= Ki+2(Z)⊗Q =

Q i = 0, i = 4k + 1 for k > 0

0 otherwise

in the concordance stable range.

24.2 Block diffeomorphisms

Definition 24.1. Diffb∂(M) is a simplicial group with k-simplices given by dif-feomorphisms f : M ×∆k →M ×∆k such that f(M × σ) = M × σ and f is idon ∂M ×∆k.

For example π0 Diffb∂(M) is diffeomorphisms of M up to pseudo-isotopy.Here [f ] ∈ πh Diffb∂(M) is represented by a diffeomorphism M ×∆k → M × kthat is identity on M × ∂∆k and ∂M × ∆k up to pseudo-isotopy. So it isisomorphic to π0 Diffb∂(M ×∆k).

In the case M = Dn with n ≥ 5,

πk Diffb∂(Dn) ∼= π0 Diffb∂(Dn+k) ∼= π0 Diff∂(Dn+k) ∼= Θn+k+1

Math 272x Notes 73

is finite. So rationally Diffb∂(Dn) has contractible components and Diff∂(Dn) →Diffb∂(Dn) is a π0-isomorphism. So

Diffb∂(Dn)/Diff∂(Dn) ∼= Diffb∂(Dn)id/Diff∂(Dn)id 'Q BDiff∂(Dn)id.

24.3 Hatcher spectral sequence

This is a spectral sequence that computes πk(Diffb∂(M)/Diff∂(M)) in terms ofπq(C(M × Ip)).

Definition 24.2. Let Dk(M) be the diffeomorphisms of M × Ik rel ∂M × Ikpreserving projections to ∂Ik on M × ∂Ik, quotient out by diffeomorphisms ofM × Ik rel ∂M × Ik preserving the projection to Ik.

There is a mapπiD

k(M)→ πi−1Dk+1(M)

because a map from (Ii, ∂Ii) to diffeomorphisms of M × Ik can be thought ofas a map from (Ii−1, ∂Ii−1) to diffeomorphisms of M × Ik × I. There is also amap

π0Dk(M)→ πk(Diffb∂(M)/Diff∂(M)).

They give a filtration by

Fiπk(Diffb∂(M)/Diff∂(M)) = im

(πiD

k−i(M)→ π0Dk(M)

→ πk(Diffb∂(M)/Diff∂(M))

).

Definition 24.3. Ck(M) is the diffeomorphisms of M × Ik × I rel ∂M × Ik × Ipreserving projection to Ik ×0 and ∂Ik × I, quotient out by diffeomorphismsof M × Ik × I rel ∂M × Ik × I preserving projection to Ik × I.

Then we have fiber sequences Dk+1(M) → Ck(M) → Dk(M) and so theirhomotopy groups give an exact couple⊕

π∗Dk(M)

⊕π∗D

k(M))

⊕π∗C

k(M)).

k

−1

jk

This spectral sequence converges to πp+q+1(Diffb∂(M)/Diff∂(M)). The E1 pageis just π∗C

k(M) and the d1-differential is π∗C(M × Ik) → π∗C(M × Ik−1)coming from restriction to M × Ik ×1. Here, C(M × Ik)→ Ck(M) is a weakequivalence.

Proposition 24.4. There is a spectral sequence

E1p,q = πq(C(M × Ip)) =⇒ πp+q+1(Diffb∂(M)/Diff∂(M))

with d1 induced by restriction to M × Ik × 1.

Math 272x Notes 74

There is a “stacking operation” on πq(C(M × Ip)) that induces a groupstructure on π∗(C(M)). There is also the standard group structure on homotopygroups. By some Eckmann–Hilton argument, they are both commutative an areequal.

There is also a stabilization C(M × Ip−1) → C(M × Ip), and it is going tosatisfy

d1[σ(g)] 7→ [g] + [g]

where g is the flip of g.

Lemma 24.5. [σ(g)] = −[σ(g)].

Proof. You can verify it with a picture.

24.4 Farrel–Hsiang theorem

IfM = Dn, only consider homotopy in the concordance stable range π∗(C(Dn))⊗Q ∼= π∗+2 WhDiff(∗)⊗Q.

The d1 differential is going to be [σ(g)] 7→ [g] + [g].

Lemma 24.6 (Farrel–Hsiang). Involution acts on π∗(C(Dn))×Q as (−1)n.

So d1 : E1,q → E0,q acts as 0 if n is odd and acts as 2 if n is oven. So inthe E2 of the Hatcher spectral sequence, everything vanishes of n is even andeverything except for p = 0 vanishes if n if odd.

Theorem 24.7. π∗(BDiff∂(Dn)) ⊗ Q ∼=

K∗+1(Z)⊗Q n odd

0 n even.in the con-

cordance stable range.

Math 272x Notes 75

25 October 30, 2017

The goal of this week is to proof the following result:

Θn ×BDiff∂(Dn) ' Ωn Top(n)/O(n)

if n 6= 4. We will prove this for n ≥ 6.

25.1 Bundles

A principal G-bundle is a space E with a free G-action with π : E → E/G =B being locally trivial. If B are nice (e.g., paracompact), these are classified byBG:

[B,BG] ←→

principal G-bundlesover B

.

The corresponding map is given by pulling back EG→ BG.Give a G-space Y we can form an associated bundle to a principal bundle

π : E → B given byπ : E ×G Y → B,

and the transition functions take values in G.If the action of G on Y is faithful, then a locally trivial Y -bundle admit-

ting transition functions in G comes up to isomorphism from from the uniqueprincipal G-bundle.

We take G = Diff∂(M) acting faithfully on M . Then we have a correspon-dence

principal Diff∂(M)-bundlesover B

←→

M -bundles with transition

functions in Diff∂(M) over B

.

If B is a smooth manifold, it is not a priori true that the total space of a M -bundle with transition functions in Diff∂(M) is also a smooth manifold. Thatis, given f : Ui ∩ Uj → Diff∂(M), it is not true that

(Ui ∩ Uj)×M → (Ui ∩ Uj)×M ; (u, x) 7→ (u, f(u)x)

is smooth. But the smoothing techniques imply that up to isomorphism we canassume transition functions have smooth adjoints. So E has a canonical smoothstructure such that π : E → B is smooth.

Definition 25.1. A smooth M-bundle π : E → B over a smooth manifold Bis a smooth map which admits bundle charts and diffeomorphisms π−1(U) →U ×M .

So if B is smooth, there is a bijection between

[B,BDiff∂(M)] ←→

smooth M -bundlesover B

/isomorphisms.

Math 272x Notes 76

25.2 Submersions

Definition 25.2. A smooth map π : E → B is a smooth submersion if ithas submersion charts. For each e ∈ E, there exists an open U ⊆ E containinge and an open V ⊆ π−1(π(e)) with diffeomorphisms

U π(U)× V

π(U).

∼=

You can also generalize this definition to a submersion chart for any K ⊆π−1(b). Then the definition is that for any open U ⊆ E open containing K,there exists an open V ⊆ π−1(b) containing K with a diffeomorphism

U π(U)× V

π(U).

∼=

Lemma 25.3. Let π : E → B be a smooth submersion. For K0,K1 ⊆ π−1(b)compact admitting submersion charts, then K0 ∪K1 also admits a submersionchart.

Corollary 25.4. If π : E → B is a proper smooth submersion, then everyπ−1(b) admits a submersion chart.

Corollary 25.5 (Ehresmann fibration theorem). For a proper submersion π :E → B and B path-connected, E → B is actually a smooth manifold bundle.

25.3 Kirby–Siebenmann bundle theorem

In general, we can’t drop the properness assumption.

Example 25.6. Consider E = R q (0,∞) and B = R. This is a smoothsubmersion but not a manifold bundle.

Theorem 25.7 (Kirby–Siebenmann). If π : E → B is a smooth submersionwhich is a topological manifold bundle, then it is a smooth manifold bundle ifdim fiber ≥ 6.

Definition 25.8. π : E → B is a topological manifold bundle if it hastopological bundle charts: for all b ∈ B there exists U ⊆ B containing B andhomeomorphisms π−1(U)→ U × π−1(b).

Today we are going to look at a weak version of this. We are going to assumethat E is homeomorphic to B ×M × R. Here is the strategy. We are going to

Math 272x Notes 77

factor

E E

B

π

Z-cover

π

and then π is going to be a proper map. Then it is a proper smooth submersionand so a smooth manifold bundle. Then π : E → B is also going to be a smoothmanifold bundle, by pulling back the smooth structure.

The input is π : E → B a smooth submersion and a continuous functionp : E → R. Let

Fb = π−1(b), Fb(m,n) = Fb ∩ p−1([m,n]).

Definition 25.9. π satisfies the fiberwise engulfing condition if for all b ∈ Band integers m ≤ n, there is a smooth isotopy ht of Fb such that

h0 = id, inth1(Fb(−∞,m)) ⊇ Fb(−∞, n)

and is compactly supported in Fb(m− 1, n+ 1).

We are going to first prove a global version.

Proposition 25.10. Suppose B is compact of dimension k. If π satisfies thefiberwise engulfing condition, then for all integers m ≤ n there is a smoothisotopy gt of E over B, such that

g0 = id, int(g1(p−1(∞,m])) ⊇ p−1(−∞, n])

and is compactly supported in p−1(m− k − 1, n+ k + 1).

These are special cases of E(r, c, C) for r ∈ (0,∞], c ∈ (0,∞), C ⊆ Bcompact: for all m ≤ n such that [m − c, n + c] ⊆ [−r + 2, r − 2] there is asmooth isotopy gt of E over B such that

g0 = id, int(g1(p−1(−∞,m]))) ⊇ p−1(−∞, n] ∩ π−1(C)

and is compactly supported in p−1(m− c, n+ c). Here are some properties:

• E(r, c, C) implies E(s, d,D) if r ≥ s and c ≤ d and C ⊇ D.

• if E(ri, c, C) for ri →∞ then E(∞, c, C).

• E(r, c, C) and E(r, d,D) implies E(r, c + d,C ∪ D) because we can dothe isotopy one after another. If C ∩ D = ∅ we can improve it toE(r,max(c, d), C ∪D).

Lemma 25.11. Fix r, and assume the fiberwise engulfing condition. For eachb ∈ B, assume that there exists an open Wb ⊆ B containing b such that for allcompact Cb ⊆Wb, E(r, 1, Cb) is true.

Math 272x Notes 78

Proof. Since π is a smooth submersion, there is a submersion chart for Fb(−r, r):

E ⊇ Ub Wb × Vb ⊆ π−1(b)

Wb.

Use this to extend hf we get for Fb to a neighborhood of b ∈Wb.

Proof of Proposition 25.10. B has a cover by k + 1 compact subset Ci, each ofwhich a finite disjoint of Cij . todotodo

Then for all i, j, we have E(r, 1, Cij and this implies E(r, 1, Ci) fora ll i. ThenE(r, k + 1, B) and so E(∞, k + 1, B).

Corollary 25.12. For every integers m ≤ n, there exists an open Emn ⊆ Econtaining p−1([m,n]) such that π|Emn : Emn → B which is a smooth manifoldbundle.

Math 272x Notes 79

26 November 1, 2017

Today we are going to finish the weak bundle theorem. Last time we provedthe following.

Proposition 26.1. For a smooth submersion π : E → B with compact Band continuous p : E → R satisfying the fiberwise engulfing condition: all fiberFb = π−1(b) have the property that, for all m ≤ n, there exists a smooth isotopygt of Fb compactly supported in Fb(m− 1, n+ 1) = Fb ∩ p−1([m− 1, n+ 1]) suchthat g0 = id and

int(g1(Fb(−∞,m))) ⊇ Fb(−∞, n).

Then a global version is also true: for all m ≤ n there exists a smooth isotopy htcompactly supported in p−1([m−dim(B)−1, n+ dim(B) + 1]) such that h0 = idand

int(h1(p−1(∞,m]))) ⊇ p−1(−∞, n]).

Corollary 26.2. For all m ≤ n, there exists an open Emn ⊆ E containingp−1([m,n]) such that π|Emn : Emn → B is a smooth manifold bundle.

Proof. Let Zmn = h1(π−1(−∞,m])) \ p−1(−∞,m) and Emn =⋃i h

i1(Zmn).

Then Emn → B factors as Emn → Emn/Z→ B and Emn/Z is a proper smoothsubmersion and hence a smooth manifold bundle. So π|Emn is a smooth manifoldbundle.

26.1 Verifying the fiberwise engulfing condition

Proposition 26.3. If Fb is homeomorphic to M × R where M is compact ofdim ≥ 5 and p = π2, then Fb satisfies the fiberwise engulfing condition.

Proof. This uses the end theorem, which answer the question “when is a non-compact manifold N the interior of a compact manifold with boundary N?

Theorem 26.4. Let N be a Cat-manifold (Cat is either Diff or Top) withoutboundary and dimN ≥ 6. Then N is an interior of compact Cat-manifoldwith boundary if and only if N has tame ends and for all ends ε a finitenessobstruction σ(ε) ∈ K0(Z[π∞1 (ε)]) vanishes.

Moreover, N is unique up to h-cobordisms starting at ∂N .

This tameness and obstruction is not dependent on the category Cat. Alsothe uniqueness part includes an important claim.

Lemma 26.5. If N is compact and W is an h-cobordism starting at ∂N , thenint(N) ∼= int(N ∪∂N W ).

Proof. We use the Eilenberg swindle. We have

int(N) ∼= N ∪ (W ∪W−1) ∪ (W ∪W−1) ∪ · · ·∼= N ∪W ∪ (W−1 ∪W ) ∪ · · · ∼= int(N ∪W ).

Math 272x Notes 80

Proof of Proposition 26.3. For m ≤ n consider Q = int(Fb(m− 1, n+ 1)). Thisis homeomorphic to M × (m− 1, n+ 1) = int(M × [m− 1, n+ 1]) and so by theend theorem Q has tame ends with vanishes finite obstructions. So Q = int(Q)and without loos of generality we may assume Q ∼= M ′ × [m − 1, n + 1] byattaching h-cobordisms.

Now note that p−1([m,n]) is a compact subset of interior of M ′× [m− 1n+1]. Then it is contained in M ′ × [m − 1 + ε, n + 1 − ε]. (Now p is not theprojection.) Use a smooth isotopy of [m − 1, n + 1] pulling [m − 1,m + 1 − ε]over [m− 1 + ε, n+ 1− ε] with compact support in [m− 1, n+ 1] and productwith idM ′ to get ht verifying fiberwise engulfing condition.

26.2 Proof of Kirby–Siebenmann bundle theorem

Theorem 26.6. If π : E → B is a smooth submersion that is a topologicalmanifold bundle with fibers of dimension ≥ 6, then it is a smooth manifoldbundle.

This is true if dim 6= 4, but definitely false in dim = 4. Freedman has acounterexample with non-diffeomorphic fibers.

Proof. Without loss of generality assume B = Dk. Then E is homeomorphic toN ×Dk over Dk. Exhaust N by compact submanifolds Ni with boundary ∂Niand take disjoint collars ∂Ni × R ⊆ N .

E N ×Dk

Dk

ϕ

Let Ei = ϕ−1(∂Ni × R). Then π|Ei is a smooth submersion whose fibers arehomeomorphic to ∂Ni × R.

By the proposition, Ei contains an open subset Ei containing ϕ−1(∂Ni ×[−1, 1]) so that π|Ei : Ei → B is a smooth manifold bundle. This is necessarily

trivial and so we get diffeomorphisms Ei ∼= Ui ×B over B.Consider smooth p|Ui×0 near 0 such that 0 is a regular value and set

Ai = p−1(0) ⊆ Ui × 0. Then Ai × B is a smooth manifold bundle which endsat a compact subset Di ⊆ E. So we get a exhaustion of E by compact subsetsDi whose boundaries are smooth manifold bundles.

Apply the Ehresmann fibration theorem to Di \ int(Di) so that each of the(Di\int(Di))\Fi are Trivializing and gluing these subbundles show that E → D finishfinishis a smooth manifold bundle.

Math 272x Notes 81

26.3 Space of smooth structures

Definition 26.7. Let U be a topological manifold. Then Sm(U) is the simplicialset with k-simplices given by (E, π, ϕ) with

E ∆k × U

∆k

π

ϕ

and ϕ is a homeomorphism, π a smooth submersion.

There is a map Sing(Homeo(U))→ Sm(U) given a smooth structure UΣ onU . This ma is given by sending ∆k → Homeo(U) to

UΣ ×∆k U ×∆k

∆k.

The bundle theorem implies that if dimU ≥ 6 then up to diffeomorphismover ∆k, every k-simplicial is of this form for some Σ. So we get

Sm(U) '∐

concordanceclasses Σ

Sing(Homeo(U))/SDiff(UΣ).

This term is the homotopy fiber of BDiff(UΣ)→ BHomeo(U). As an example,

Sm(Rm) ' Top(m)/O(m), Sm∂(Dn) '∐

concordanceclasses Σ

BDiff∂(DnΣ).

Index

Alexander’s theorem, 31algebraic K-theory, 64

Casson trick, 58cobordism, 4, 44cocore, 44collar, 9connected sum, 25convolution, 15core, 44critical point, 32critical value, 32

exponential map, 12

fiberwise engulfing condition, 77finger move, 58

Γ-space, 60geodesic, 12

h-cobordism, 4, 48handle decomposition, 43Hatcher’s theorem, 31Hatcher–Wagoner–Igusa sequence,

69Hessian, 38

index, 38, 44isotopy, 21

jet, 7jet transversality, 36

K-group, 60, 61Kirby–Siebenmann bundle

theorem, 76

level set, 41

manifoldCr-manifold, 6topological manifold, 6

Morse function, 38generic, 41Morse lemma, 39

nondegenerate critical point, 38

partition of unity, 9principal G-bundle, 75pseudo-isotopy, 70

regular point, 32regular value, 32retractive finite space, 64

Sard’s lemma, 32signature, 53smooth M -bundle, 75smooth structures, 81smooth submersion, 76stratification, 37surgery, 44

topological manifold bundle, 76transversality, 33transverse sphere, 44tubular neighborhood, 13

vector bundle, 60

Waldhausen category, 63Whitehead torsion, 52Whitney embedding, 15Whitney topology, 6Whitney trick, 58

82