Pseudo-Isotopies of Compact Manifolds

Allen Hatcher and John Wagoner

1973

Contents

0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I Pseudo-Isotopies of Non-simply connected manifolds and the functor K2. 20.2 Introduction and Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Pseudo-isotopies and real-valued functions 61.1 Cerf’s “functional approach” to pseudo-isotopies. . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The Stratification of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Gradient-like vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 “Nice” families of gradient-like vector fields. . . . . . . . . . . . . . . . . . . . . . . . . 18

0.1 Preface

The two papers in this volume compute the components of the space of pseudo-isotopies of a compactmanifold of dimension at least seven and the main result can be viewed as a third step in relating differentialtopology to algebraic K-theory. Historically, the first step was Whitehead’s theory of simple homotopy types,the Franz-Reidemeister torsion invariant, and then later Smale’s h-cobordism theorem and its generalizationto the non-simply connect case; namely, the s-cobordism theorem of Barden-Mazur-Stallings, which showedhow the Whitehead group measured the obstruction to putting a product structure on an h-cobordism.Next, work of Browder-Levine-Livesay followed by work of Siebenmann, of Golo, and of Wall in the non-simply connected case showed how the Grothendieck group K0 of the category of finitely generated, projectivemodules gave the obstructions to putting a boundary on an open manifold. On the algebraic side Serre showedthat algebraic vector bundles over an affine variety correspond to finitely generated projective modules overits coordinate ring. The Swan showed that the Atiyah-Hirzebruch group of virtual vector bundles over acompact space was just K0 for the ring of continuous functions on that space. Bass studied the functorK1 on rings, of which the Whitehead group is a suitable quitient, and showed how to fit K0 and K1 intoan exact sequence similar to the one in the ATiyah-Hirzebruch K-theory. Consequently, a feeling emergedthat there must be an “algebraic” K-theory concerned with an appropriate sequence of functors K0, K1,K2, etc. Such a theory has recently been developed and is an active area of research. The third step beganon the geometric side with Cerf’s theorem that pseudo-isotopy implies isotopy in the simply connected casein dimensions at least five. Just as the s-cobordism theorem was related to the uniqueness of putting aboundary on an open manifold, the pseudo-isotopy problem measures the uniqueness of a product structureon a trivial h-corbordism and it seemed natural that a non-simply connected version of Cerf’s result wouldbe related to a functor K2. Around 1967 Milnor defined a K2 group along the lines of Steinberg’s workson universal coverings of Vhevalley groups and this turned out to be what was needed. However, unlikethe previous two geoemtric problems corresponding to K0 and K1 the non-simply connected pseduo-isotopytheorem requires a second obstruction which depends not only on the fundamental group but on the secondhomotopy group as well. For a precise statement of the restul see the Introduction to Part I of this volume.

1

Part I

Pseudo-Isotopies of Non-simplyconnected manifolds and the functor

K2.

2

0.2 Introduction and Summary of results

Let (M,∂M) be a smooth, compact, C∞ manifold of dimension n. A pseudo-isotopy of (M,∂M) is adiffeomorphism f : (M,∂M) × I → (M,∂M) × I such that f restricted to M × 0 is the identity and frestricted to M × I is an isotopy (i.e. it preserves projection onto I). Let P = P(M,∂M) denote the groupof pseudo-isotopies of (M,∂M) where the multiplication is composition and we give P the C∞ topology.The problem is to compute π0(P). See the introduction to [?] and sections 4 and 5 of [?] for applications ofsuch a computation.

Here are the algebraic K-theory functors we will need:Following [?] let Λ be any associate ring with unit and define the Steinberg group St(Λ) to be the free

group generated by symbols xij(λ) where 1 ≤ i, j <∞, i 6= j, and λ ∈ Λ, modulo the relations

1. xij(λ) · xij(µ) = xij(λ+ µ)

2. [xij(λ), xkl(µ)] = 1 for i 6= l and j 6= k

3. [[xij(λ), xkl(µ)] = xik(λµ) for i, j, k distinct.

Sometimes we write xλij for xij(λ). Let GL(Λ) = limn→∞GLn(Λ) be the infinite general linear group andE(Λ) ⊂ GL(Λ) be the subgroup generated by the elementary matrices eλij , where eλij is the identity on thediagonal, has λ as the (i, j)th entry, and is zero elsewhere.

The correspondence xλij → eλij defines a surjective homomorphism

π : St(Λ)→ E(Λ)

and Milnor defines the functor K2 in [?] as

K2(Λ) = ker(π).

The group K2 is abelian because it is the center of St(Λ). See [?]. Now let Λ = Z[π1M ], the integralgroup ring of π1M . Let W (±π1) ⊂ St(Λ) denote the subgroup generated by words wij(±g) of the formxij(±g) · xji(±g−1) for g ∈ π1M . Let W0(±π1) = K2(Λ) ∩W (±π1) and define

Wh2(π1M) = K2(Z[π1M ]) mod W0(±π1M).

This is the first obstruction group for measuring π0(P).To define the second part let (Z2 × π2M)[π1M ] denote the group of all functions f : π1M → Z2 × π2M

which are zero except on finitely many elements of π1M ; that is, (Z2 × π2M)[π1M ] is the direct sum of|π1M | many copies of Z2 × π2M . Any element of (Z2 × π2M)[π1M ] can be written as a finite formal sum∑i αiσi where αi ∈ Z2 × π2M and σi ∈ π1M . Let π1M act tirivially on Z2 and let it act in the usual way

on π2M . If α ∈ Z2 × π2M and τ ∈ π1M , denote the action of τ on α as ατ . Define

Wh1(π1M ; Z2 × π2M)

to be (Z2 × π2M)[π1M ] modulo the subgroup generated by α · σ−ατ · τστ−1 and β · 1 for α, β ∈ Z2 × π2Mand σ, τ ∈ π1M . Here 1 denots the identity of π1M . See [?] for a more conceptual definition of this group.

The main result is

Theorem. For any connected, compact, C∞ manifold (Mn, ∂Mn) there are homomorphisms

Σ : π0(P)→Wh2(π1M)

andθ : π0(P)→Wh1(π1M ; Z2 × π2M)

such that both are surjective for n ≥ 5 and whenever n ≥ 71

Σ + θ : π0(P)→Wh2(π1M)⊕Wh1(π1M ; Z2 × π2M)

is an isomorphism.1In fact, n ≥ 6 is sufficient

3

The homomorphism Σ was constructed and its kernel identified geometrically by both of the authorsworking independently. See [?, ?]. The homomorphism θ was constructed by the first author in [?], whichis Part II of this volume. In Part I we shall prove the main theorem except for giving the construction of θ.This theorem has also been announced by I.A. Volodin in [?].

When π1M = 0 the group Wh2(π1M) vanishes because K2(Z) ∼= Z2 with the generator being w12(1)4.See [?]. Also Wh1(1; Z2 × π2M) vanishes as one sees directly from the definition above. Although in thegeneral case Σ + θ is injective for n ≥ 7, in the simply connected case our methods work when n ≥ 5 torecover Cerf’s theorem [?] that π0(P) = 0.

Here is some information presently known about Wh2(π).

1. If π is finite, then Wh2(π) is probably finite. See [?, ?].

2. Wh2(Z20) has at least order 5 (Milnor).

3. Wh2(π × Z) = Wh2(π)⊕Wh1(π)⊕ (?)Here Wh1(π) is the usual Whitehead group. See [?]. This algebraic result was suggested by geometricexamples in [?].

4. Wh2(free abelian group) = 0. This follows from (3) and the fact recently proved by Quillen that fora left regular ring A there are isomorphisms K2(A) ∼= K2(A[t]) and K2(A[t, t−1]) ∼= K2(A) ⊕K1(A).Compare [?], [?, Chap. XII], or [?].

5. Wh2(free group) = 0 (Swan and Gersten using methods of Quillen).

The formula in (3) is related to pseudo-isotopies on M × S1 where π = π1M . Using geometric argumentWu-chung Hsiang has recently in [?] given a description of (?), showing in particular that (?) is not finitelygenerated for π = Zp2 ×Z3 where p is an odd prime. Pseudo-isotopies on a manifold which is the connectedsum of Xn and Y n with π1X = A and π1Y = B are related to the computation of Wh2(A ∗ B). In [?] itwas shown that Wh1(A ∗ b) = Wh1(A)⊕Wh1(B). Is the same true for Wh2? Note that Wh1(π; Z2 × π2)behaves badle with respect to connected sum. For example, when π1X = π1Y = Z2 and π2X = π2Y = 0 wehave Wh1(Z2; Z2) = Z2 while Wh1(Z2 ∗ Z2; Z2) is not finitely generated.

Beyond this volume there is the problem of computing the higher homotopy groups πk(P) for k ≥ 1.The techniques used here and those of [?] indicate that these groups will probably depend more and moreon the tangential homotopy type of M as k gets large. Part of πk(P) should however depend only on π1Mand there should be higher algebraic K-theory functors Whk+2(π1M) together with surjections πk(P) →Whk+2(π1M). Compare [?, ?, ?]. One problem in studying πk(P) with Cerf’s approach [?] of using thestratification of the space of smooth real valued functions on M × I is that continuous moduli appear insmooth singularities of high codimension. However, Mather’s work on singularities shows there ar eonlyfinitely many singularity types up to piecewise lienar equivalence in a given codimension. Thus maybe thepiecewise-linear case is easier to handle. Coincidentally Burghelea and Lashof have shown recently that thespace Pp.l. of piecewise-linear pseudo-isotopies has the same number of components as the space Pdiff ofsmooth pseudo-isotopies. However, the map π1Pdiff → π1Pp.l. is not an isomorphism (n large). See [?, ?].

Here is how Part I is organized: In Chapter I we explain Cerf’s approach to the pseudo-isotopy problemusing one parameter families in the space F of C∞ functions on M × I and then introduce the space ofgradient-like vector fields. The key concepts are the graphic of a k-parameter family, the startification ofF ,nice gradient-like families of vector fields, i/j intersections of trajectories, general position of a family ofgradientlike vector fields, independence of trajectories, suspensions, and one and two parameter ordering.See the tables “F 1 graphics” and “F 2 graphics” in §2 for a summary of how one and two parameter familiesof functions behave. Section 8 discusses one and two parameter ordering and shows how to deform one andtwo parameter families using essentially only general position methods into families with a graphic that isrelatively simple, i.e. ordered. In later parts of the paper we usually just start with ordered families.

Chapter II shows how the geoemtry of the i/i corssings in certain one-parameter families of gradient-likevector fields gives rise to a word in the Steinberg group. In order to show this word determines a well-defined invariant in Wh2(π1M) it is necessary to see what happens as the one-parameter family is deformed.The reader should consult Table 2.3 in §2 of II for a summary of the three basic tpes of changes in thegraphic which must be analyzed. Chapter III develops the algebraic machinery used in proving that the

4

Steinberg word of a one parameter family gives a well-defined element in Wh2(π1M). The amterial is theone-parameter analogue of what is done in defining the Whitehead torsion of an acyclic complex of lengthgreater than two.

Chapter IV completes the definition of the Wh2(π1M) invariant of a pseduo-isotopy. The main work isto show why the geometric vhanges which occur when a one-parameter family of gradient-like vector fields isdeformed only alter the Steinberg word of that one-parameter family by relations defining the Wh2 group.

Chapter V is mostly geometryc. Techniques for simplifying the graphic of a k-parameter family are givenand in paritcular it is shown that for 0 ≤ k ≤ 2 and k-parameter family can be reduced to one with criticalpoints having indices only in two consecutive dimensions. This is needed in Part II for the definition of theWh1(π1M ; Z2×π2M) invariant of a pseudo-isotopy. Attention is called to the alst section which shows howthe definition of the Wh2(π1M) invariant is much simpler in the “two-index” situation.

Chapter VI and VII give the proof of the main result computing π0(P) except for showing that thesecond obstruction in Wh1(π1M ; Z2×π2M) is well-defined. This past part is computed in [?], which is PartII of this volume. Chapter VIII gives product and duality formulae for the Wh2 invariant of a pseudo-isotopy.

In addition to constructing the Wh1(π1; Z2×π2) invariant, Part II includes product and duality formulaefor the second obstruction.

5

Chapter 1

Pseudo-isotopies and real-valuedfunctions

In this chapter we begin by recalling Cerf’s reduction of the pseudo-isotopy problem to the study of thespace of all C∞ functions on M × I. Then we discuss a number of results and techniques which form thegroundwork for the rest of the paper.

1.1 Cerf’s “functional approach” to pseudo-isotopies.

Let F be the space of C∞ functions f : M × I → I such tath f(x, 0) = 0 and f(x, 1) = 1 for all x ∈ M , fhas no critical points near M × 0 and M × 1, and f(x, t) = t for all x ∈ ∂M . Here I denotes the interval[0, 1]. Let p : M × I → I denote the standard projection. Let E ⊂ F be the subspace consisting of thosefunctions with no critical points. The correspondence g → p ◦ g induces a fibration

I →P →π

E

where the fiber I = π−1(p) is just the space of isotopies of the indetity of M (i.e. the space of level preservingdiffeomorphisms of M ×I which are the identity on M ×0). To see, for example, that π is onto choose f ∈ Eand choose a Riemannian metric on M . Give M × I the product metric. A diffeomorphism g of M × I toitself with p ◦ g = f is obtained by mapping the interval x × I to the trajectory of grad f which starts atx× 0 ∈M × 0 and ends somewhere in M × 1. Now the space I is contractible because it is just the spaceof all paths from the identity in Diff(M,∂M). Hence there is a homotopy equivalence

π : P → E .

Since F is contractible we haveπi(P) ∼= πi(E ) ∼= πi+1(F ,E ; p).

The general plan, then, for measuring the obstruction to connecting any pseudo-isotopy g ∈Pto the identityby a path in P is to join p to p ◦ g by a path in F and then try to deform this path down into E keepingendpoints fixed.

Since π0(P) is a group the bijection above induces a group structure on π0(E ) and π1(F ,E ; p). Here ishow to do this directly. Let f, g ∈ F . Deform f and g by a very small amount (so that if f and g are inE they remain in E ) until they agree with p on M × [0, ε] and M × [1− ε, 1] for small ε > 0. Then definedf#g : M × I → I by

f#g(x, t) =

{12f(x, 2t) 0 ≤ t ≤ 1

212g(x, 2t− 1) + 1

212 ≤ t ≤ 1

If [f ] and [g] are in π0(E ) then[f ] · [g] = [f#g].

6

Similarly, if [fS ] and [gS ] are in π1(F ,E ; p) are represented by paths fS and gS , 0 ≤ s ≤ 1, then

[fS ] · [gS ] = [fS#gS ]

.

Lemma 1.1. If dimM ≥ 6, then π0(P) is abelian.

This lemma will not be needed in the sequel and in fact for dimM ≥ 7 it is a consequence of the maintheorem.

Proof. Choose an ordered morse function f : M → R (i.e. an index p > index q =⇒ f(p) > f(q)). Let3 ≤ k ≤ dimM−3 be a fixed integer and let c ∈ R be a non-critical value of f such that for any critical pointp of f , f(p) < c iff index p ≤ k and f() > c iff index p > k. Let A = f−1((−∞, c]) and B = f−1([c,∞)).Both A and B have handle decompositions in which each handle has codimension at least three. Now letF and G ∈ P. Use the fact that “pseudo-isotopy implies isotopy” in codimension at least three [?] toinductively deform F on the subspaces (handle of A) × I until it becomes the identtiy on A × I and hassupport in B × I. See [?]. Similarly deform G so that it is the identity on B × I and has support in A× I.Then clearly F ·G = G · F .

Remark. Since the space of paths from the identity in Diff(∂M) is contractible the space P(M,∂M) definedin the introduction has the same homotopy type as the space of diffeomorphisms of (M,∂M)× I which arethe identity on M × 0 and ∂M × I. We shall henceforth identify these spaces.

1.2 The Stratification of F .

In this section we recall from [?] some facts about the low dimensional strata in the space of all smooth realvalued functions on a manifold. We shall describe what a “generic” k-parameter family of maps looks likefor 0 ≤ k ≤ 2.

Let V n+1 be a smooth compact manifold with ∂V = C ∪D. Let F denote the space of all C∞ functionsf : (V ;C,D)→ (I : 0, 1) with no critical points near ∂V . As in [?] we can write F as the disjoint union

F = F 0 ∪F 1 ∪F 2 ∪F 3 ∪H

where F k consists of those functions of codimension k (0 ≤ k ≤ 3) and H consists of functions of highercodimension. For 0 ≤ k ≤ 3 we can compute the codimension of a function as follows: Let f ∈ F and letp be an isolated critical point of f . To codimension of p is the codimension (as a vector space over the realnumbers) of the ideal generated by the partial derivatives of f in the ideal of all germs of functions on Vvanishing at p. The codimension of a critical value α of f is the number of critical points in f−1(α) minusone. Let

ν1(f) = sum of codimensions of critical points

ν2(f) = sum of codimensions of critical values

Then for 0 ≤ k ≤ 3codimension of f = ν1(f) + ν2(f).

The canonical forms for critical points of codimension less than or equal to two areCodim 0 (non degenerate critical point)

(0) −x21 − ...− x2

i + x2i+1 + ...+ x2

n + x2n+1

Codim 1 (birth or death point)

(1) −x21 − ...− x2

i + x2i+1 + x2

n + x3n+1

Codim 2 (dovetail point)

(2) −x21 − ...i− x2

i + x2i+1 + ...+ x2

n ± x4n+1.

7

We shall say for the above models that 0 is a critical point of index i in the cases (0) and (1). In the case(2) we say 0 is a critical point of index i when “+x4

n+1” is the last term and of index i + 1 when “−x4n+1”

is the last term.The canonical models for the unversal unfoldings of these singularities are respectively

(0’) −x21 − ...− x2

i + x2i+1 + ...+ x2

n+1

(1’) −x21 − ...− x2

i + x2i+1 + ...+ x2

n + txn+1 + x3n+1

(2’) −x21 − ...− x2

i + x2i+1 + ...+ x2

n ± (txn+1 + sx2n−1 + x4

n+1).

For 0 ≤ k ≤ 3 the subspace F 0 ∪ ... ∪ F k is open in F and for any f ∈ F k the stratification islocally trivial at f . This means that there is a neighborhood of f in F of the form Rk ×W where W isa neighborhood of f in F k and where there is a stratification of Rk with 0 as a point stratum such thatthe stratification induced on Rk ×W by the F i is just the product stratification whose strata are (strataof Rk)×W with 0 ×W = W . It is an interesting open problem to find a good stratification of H . Recentexamples of H. Hendriks (to appear in Comptes Rendus) show that the stratification of H by codimensionis in general not locally trivial above dimension 7.

The following is an explicit description of F 0, F 1, and F 2.The stratum F 0.

We must have ν1(f) = ν2(f) = 0. Hence F 0 consists of functions with only non-degenerate criticalpoints and distinct critical values.The stratum F 1 = F 1

α ∪F 1β .

F 1α: ν1(f) = 1 and ν2(f) = 0. There is just one birth point, all other critical points are non-degenerate,

and the critical values are distinct.

F 1β : ν1(f) = 0 and ν2(f) = 1. All critical points are non-degenerate and there is exactly one pair of critical

points with the same critical value.

The stratum F 2. There are six types of function in F 2:

F 2α: ν2(f) = 2 and ν1(f) = 0. There is exactly one dovetail point and all critical values are distinct.

F 2β : ν2(f) = 2 and ν1(f) = 0. There are exactly two birth points and all critical values are distinct.

F 2γ : ν1(f) = 1 = ν2(f). There is one birth point and one double critical value for two non-degenerate

points.

F 2δ : ν1(f) = 1 = ν2(f). A birth point and a non-degenerate point have the same critical value.

F 2ε : ν1(f) = 0 and ν2(f) = 2. Three non-degenerate points have the same critical value.

F 2ζ : ν1(f) = 0 and ν2(f) = 2. There are two double critical points.

If fz : V n+1 → R is a k-parameter family where z varies over a parameter domain D ⊂ Rk define thegraphic of the family fz to be ⋃

z∈D[critical values of fz].

The graphic is a subset of D×R.i+1

i. Here the i+1 and the i appearing next to the lines in graphic

indiate that those lines are the images of critical points of index i+1 and i respectively. When n = 0 in (1’) the

8

actual one parameter family looks like

t<0 t=0 t>0

The graphic of the one parameter family

is

1

1

0

.

Another typical graphic which might occur for a one parameter family is

i+1

ii

i+1

j+1

j

The graphic of the two-parameter family ft,s = −(tx+ sx2 + x4) which is the unversal unfolding of x4 isa subset of R2 × R. The intersections of this graphic with the planes s =constant are

1

1

0

1

s>0 s<0

For a fixed s < 0, the one parameter family ft,s is

9

Here is a table of some universal unfoldings of the functions in F 1 and F 2:

F 1 graphics

F 1α :

i+1

i

Death Point

or

i+1

iBirth Point

F 1β :

i

j crossingpoint

F 2 graphics

F 2α:

i+1 i+1

i

i+1

(index i+1) or

i+1

i i

i+1

(index i)

F 2β :

i+1

i

j+1

j

F 2γ :

10

F 2δ :

F 2ε :

F 2ζ :

Any k-parameter family fz : V n+1 → R where z ∈ Dk determines a map F : Dk×V → Dk×R preservingprojection onto Dk where F (z, x) = (z, fz(x)). It also determines a map α : Dk → F where α(z) = fz. Thefollowing are equivalent statements for 0 ≤ k ≤ 2.

(a) α(Dk) ⊂ F 0 ∪F 1 ∪F 2 and the map is transverse to each stratum F i.

(b) The map F is “generic”. This means first that F has only transverse singularities of type Σn+1,0,Σn+1,1,0, Σn+1,1,1,0 (c.f.[?]). Furthermore let Σ ⊂ Dk × V denote the set of all singular points of F . Σis a smooth k-dimensional submanifold ofDk×V and Σ = Σ0∪Σ1∪Σ2 where Σ0 = Σn+1,0(F ) =singularpoints of type Σn+1,0, Σ1 = Σn+1,1,0(F ), and Σ2 = Σn+1,1,1,0(F ). In fact Σi consists of those points(z, p) ∈ Dk × V such that p is a critical point of fz of codimension i. Σ1 ∪Σ2 is a smooth submanifoldof Σ of codimension k−1 and Σ2 is a smooth submanifold of dimension k−2. The second condition forgenericity of F requires that if Σαi and Σβj are componenets of Σi and Σj then the maps F : Σαi → Dk×Rand F : Σβj → Dk × R are in general position.

Thom transversality methods show that any k-parameter family can be approximated by a generic familyas described above when 0 ≤ k ≤ 2. Consequently, any k-parameter family can be deformed off of strata ofcodimension greater than k and

πi(F ,F 0 ∪ ... ∪F k) = 0

for i ≤ k ≤ 2.Let fz : V → R be a generic k-parameter family and let (u, p) ∈ Σ ⊂ Dk × V . A parametrized

version of the splitting theorem of [?], see [?] also, says that there is a neighborhood U of u in Dk and ak-parameter family of imbeddings ϕk : Rn1 → V with ϕu(0) = (u, p) such that for some quadratic formq(x1, ..., xn) = ±x2

1 ± ...± x2n and some k-parameter family dz : R→ R we have

(∗) f ◦ ϕ(x1, ..., xn+1) = q(x1, ..., xn) + dz(xn+1)

for z ∈ U . If (z, p) ∈ Σi, then dz has 0 as a critical point of codimension i. This says that fz is essentiallya suspension of du (see §5) and so its behavior is like that of dz near (u, p). Canonical models for genericfamilies dz : R→ R areOne parameter

D(t, x) = (t,±x2), 0 ∈ Σ0 orD(t, x) = (t, tx+ x3), 0 ∈ Σ1.

Two parametersD(t, s, x) = (t, s,±x2), 0 ∈ Σ0 or

11

D(t, s, x) = (t, s, sx+ x3), 0 ∈ Σ1 orD(t, s, x) = (t, s,±(tx+ sx2 + x4)), 0 ∈ Σ2.A complete description of F 3 could be given as was done for F 1 and F 2. We content ourselves with

listing the codimension three critical points and their universal unfoldings. See [?]. Two of these three kindsof singularities will be used to prove a result in chapter V §3 (Th. 3.1.b) which, however, is not necessaryto our proof of the main theorem. To describe these singularities it suffices by the splitting (∗) to give thedegenerate part of the function; namely, du’s. The function d0 is usually called the organizing center.

Name Organizing Center Universal unfoldingButterfly x5 t1x+ t2x

3 + t3x3 + x5

Hyperbolic umbilic x3 + y3 x3 + y3 + t1xy − t2x− t3yElliptic umbilic x3 − 3xy2 x3 − 3xy2 + t1(x2 + y2)− t2x− t3y

The trace of a map α : Dk → F is the decomposition of Dk into the disjoint sets α−1(component ofsome F k). Here are some examples.

The trace of the universal unfolding of the dovetail singularity is

F 1β

F 1α

F 2

F0

F 1α

Consider a two parameter family ft,s : V → R with a graphic like

nothing

nothing

i+2

i+1

i

i+1

i

i+1 i+1

i+2

i

i+1i+1

i+2

This has trace

F 0

F 1α

F 2α

F 1β

12

This is a two-dimensional section in the trace of the universal unfolding of the hyperbolic umbilic. See[?].

1.3 Gradient-like vector fields

We shall be interested in studying triples (η, f, µ) where f : V n+1 → R is a C∞ function on V as in §2 andη is a vector field on V that is gradient-like for f with respect to the Riemannian metric µ on V . The termgradient-like means that

1. dfx(ηx) > 0 whenever x is not a critical point of f is not a critial point of f and

2. There is a neighborhood U for each critical point p such that η(x) = gradµ f(x) for x ∈ U where thegradient is computed using µ.

Let F denote the space of such triples and, when V = M × I, let E ⊂ F denote the subspace consistingof those triples (η, f, µ) where f has no critical points. In the case ∂M 6= 0 we take F to be all those triples(η, f, µ) such that near ∂M×I the metric µ is the product metric of some fixed metric on M and the standardmetric on I. Now fix a Riemannian metric µ on V . Then the math F → F given by f 7→ (gradµ f, f, µ) isa homotopy equivalence which induces a homotopy equivalence of pairs

(F ,E )→∼= (F , E ).

The deformation retraction of F down in F is done in two stages. First deform (η, f, µ) to gradµ f, f, µ)via the path (t · gradµ f + (1− t)η, f, µ), 0 ≤ t ≤ 1, and then deform (gradµ f, f, µ) to (gradµ f, f, µ) by thepath (gradµt

f, f, µt) where µt = tµ+ (1− t)µ) for 0 ≤ t ≤ 1. We have in particular

π1(F ,E ; p) ∼= π1(F , E ; p)

where p = (gradµ p, p, µ).Now let (η, f, µ) ∈ F and let p ∈ V be an isolated critical point of f . let ϕt be the one parameter

family of diffeomorphisms generated by η. Define the stable and unstable sets of p, written W (p) and W ∗(p)respectively, by the equations

W (p) = {x ∈ V | limt→∞

ϕt(x) = p}

andW ∗(p) = {x ∈ V | lim

t→−∞ϕt(x) = p}

Let p and q be tw critical points of f of index i and j respectively. suppose f(p) > f(q) and let L = f−1(c)be an intermediate level surface where f(p) > c > f(q). Then the intersection W (p) ∩W ∗(p) ∩ L will becalled an i/j intersection.

Supposed we have a smooth k-parameter family (ηz, fz, µz) in F , z ∈ Dk, such that the map F :Dk × V → Dk × R is in general position as in §2. For each point (z, pz) in the critical set Σ, pz is a criticalpoint of fz and we have the sets W (pz) and W ∗(pz) contained as subsets of z × V . We shall need to knowhow these sets vary as z, pz moves around in Σ because the Wh2 invariant for pseudo-isotopies comes fromthe i/i intersections in a one-parameter family in F and the (Wh1(π1M ; Z2 × π2M) invariant comes fromthe i+ 1/i intersections in a one-parameter family.

In the remainder of this section we will first give six examples of the behavior of the stable and unstablesets near the generic singularities and then give an existence theorem for nice families of gradient like vectorfields. To economize in notation we shall often suppress the notation for the Riemannian metric in a k-parameter family and shorten (ηz, fz, µz) to (ηz, fz).

Example 1. Each function fz of the k-parameter family has only isolated, non-degenerate critical points, sayof index i. Then W (pz) ' Ri and W ∗(pz) ' Rn+1−i and W (pz) intersect W ∗(pz) transversely in the pointpz. As z moves smoothly in Σ0 = Σ the stable and unstable manifolds vary smoothly. This is just “stablemanifold theory”, see [?].

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Example 2. (k = 1). Consider the one-parameter family (grad ft, ft, µ) where µ is the standard metric onRn+1 and ft(x1, ..., xn1) = −x2

1− ...−x2i +x2

i+1 + ...+x2n+ txn+1 +x3

n+1. When t = 0 there is just one criticalpoint, namely 0, which lies in Σ1. When t < 0, ft has two non-degenerate critical points at = (0, ..., 0,

√−t/3)

and bt = (0, ..., 0,−√t/2) of index i and i + 1 respectively. Both at and bt are in Σ0. For t > 0 ft has no

critical points at all. Then for t < 0 we have (in Ri × Rn−i × R), where ct =√−t/3,

W (at) = Ri × 0× {ct},W ∗(at) = 0× Rn−i × {−ct < xn+1}W (bt) = Ri × 0× {xn+1 < ct}W ∗(bt) = 0× Rn−i × {ct}

For t = 0 we have

W (0) = Ri × 0× {xn1 ≤ 0} and W ∗(0) = 0× Rn−i × {0 ≤ xn+1}.

In particular W (0) and W ∗(0) are half spaces.Let ε > 0 and choose d > 0 so that for t ∈ [−ε, ε] the critical values of ft are contained in (−d, d). The

corresponding graphic is

d

-d

i+1

i

For t < 0, let Xt = f−1t ([−d, d])∩(W (at)∪W ∗(at)∪W (bt)∪W ∗(bt). Let X0 = f−1

0 ([−d, d])∩(W (0)∪W ∗(0)).For t ≤ 0, let Xt(±d) = f−1

t (±d) ∩Xt. Then for t ≤ 0 we have

1. Xt is contractible.

2. Xt(d) is an (n− i) disc with boundary the (n− i− 1) sphere W ∗(bt) ∩ f−1t (d)

3. Xt(−d) is an i-disc with boundary the (i− 1) sphere W (at) ∩ f−1t (−d).

For t < 0 we also have

4. The i-sphere W (bt) ∩ f−1t (0) and the (n− i) sphere W ∗(at) ∩ f−1

t (0) intersect each other transverselyin a single point in the level surface f−1

t (0).

Example 3. Consider the two parameter family (ηt,s, ft,s) where

ft,s = −x21 − ...− x2

i + x2i+1 + ...+ x2

n + (txn+1 + x3n+1)

and ηt,s is the gradient of ft,s with respect to the standard metric. This is just a one-parameter version ofthe previous example. In each slice s = constant the behavior of the stable and unstable manifolds is as inexample 2.

Example 4. (Dovetail singularity) Let (ηt,s, ft,s) be the two parameter family where

ft,s(x1, ..., x2n+1 = −x2

1 − ...− x2i + x2

i+1 + ...+ x2n − (txn+1 + sx2

n+1 + x4n+1)

and ηt,s is the gradient of ft,s with respect to the standard metric of Rn+1. As s goes from positive tonegative the change in the graphic is

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i+1 i+1 i+1

i

i+1

s>0 s=0 s<0

For s > 0 there is just one non-degenerate critical point of index i + 1 at each time t; for s = 0 there is anon-degenerate critical point when t 6= 0 and a codimension two critical point, namel y 0, when t = 0. Fort = s = 0, we have (in Ri × Rn−i × R) W (0) = Ri × 0× R and W ∗(0) = 0× Rn−i × 0.

Fix s < 0 and for all t set (ηt, ft) = (ηt,s, ft,s). Let at for t < δ and bt for δ < t denote the critical pointsof index i+ 1 as indicated in the diagram below. For −δ < t < δ let ct denote the critical point of index i.let c−δ and cδ be the birth and death critical points respectively.

t=0t=-δ t=δ

r

w

v

u

ft(ct)

i

i+1i+1

ft(at) ft(bt)

Here is how the stable and unstable sets vary within the intermediate level surfaces. See [?, IV.14] andparticularly [?, Chap. 2].

1. Near the birth and death points the situation is just as in Example 2 above.

2. In the u-level Kt = f−1t (u):

W ∗(at) ∩Kt∼= Sn−i−1(at)

W ∗(bt) ∩Kt∼= Sn−i−1(bt)

W ∗(ct) ∩Kt∼= Sn−i−1(bt) where

{Sn−i−1 × 0 ∼= Sn−i−1(at)Sn−i−1 × 1 ∼= Sn−i−1(bt)

. See the following diagram:

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t<-δ -δ≤t≤δ δ<t

Sn-i-1(at)Sn-i-1x I

Sn-i-1(bt)

Sn-i-1(at)

Sn-i-1(bt)

Another diagram illustrating the variation is

tSn-i-1(at)

Sn-i-1(bt)Sn-i-1 x I

3. In the v-level surface Lt = f−1t (v):

For t < 0:

W (at) ∩ Lt ∼= Si(at)W ∗(c−δ) ∩ L−δ ∼= Dn−i(c−δ)W ∗(ct) ∩ Lt ∼=

◦Dn−i(ct) for − δ < t < 0

W ∗(bt) ∩ Lt ∼= ∂(W ∗(ct) ∩ Lt) ∼= Sn−i−1(bt) for − δ < t < 0

For t > 0:

W (bt) ∩ Lt ∼= Si(bt)W ∗(cδ) ∩ Lδ ∼= Dn−i(cδ)W ∗(ct) ∩ Lt ∼=

◦Dn−i(ct) for 0 < t < δ

W ∗(at) ∩ Lt ∼= ∂(W ∗(ct) ∩ Lt) ∼= Sn−i−1(at) for 0 < t < δ

Note that for −δ ≤ t < 0, Si(at) intersects◦Dn−i(ct) transversely in exactly one point; similarly for

0 < t ≤ δ, Si(bt) intersects◦Dn−i(ct) transversely in one point. See the following diagram:

Si(at)

◦Dn−i(ct)

Sn−i−1(bt)

Si(bt)

Sn−i−1(at)

◦Dn−i(ct)

t < 0 0 < t

4. In the w-level Pt = f−1t (w): For −δ < t < δ W (at) ∩W ∗(bt) = φ and W ∗(at) ∩W (bt) = φ. Hence

W (at) ∩ Pt ' Si(at)W (bt) ∩ Pt ' Si(bt)W ∗(ct) ∩ Pt ' Sn−i(ct)

In fact each of Si(at) and Si(bt) intersect Sn−i(ct) transversely in a single point in Pt.

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5. In the r-level Qt = f−1t (r):

W (at) ∩Qt ∼= Si(at) (t < −δ)W (at) ∩Qt ∼=

◦Di(at) (−δ ≤ t < δ)

W (bt) ∩Qt ∼=◦Di(bt) (−δ < t ≤ δ)

W (ct) ∩Qt ∼= Si−1(ct) ∼= ∂◦Di(at) = ∂

◦Di(bt) (−δ < t < δ)

W (c−δ) ∩Qt ∼= Di(c−δ)W (cδ) ∩Qt ∼= Di(cδ)W (bt) ∩Qt ∼= Si(bt) (δ < t)

See the following diagram:

t < δ

Si(at)

t = −δ

◦Di(a−δ)

Di(cδ)

◦Di(bt)

◦Di(at)

◦Di(bδ)

Di(cδ)

t = δ

δ < t

Si(bt)

Si−1(ct)

Example 5. (0 ≤ k ≤ 2). Let F : Dk × R → Dk × R be a generic map where F (z, x) = (z, fz(x)) andlet µz, z ∈ Dk, be a smooth family of metrics on R. Let ηz = gradµz

fz. Since we are dealing here withgradients of functions of a single real variable the situation is easy to analyze and one sees that the stableand unstable sets of points in Σ have the same intersection phenomena as in the above examples where thestandard metric was used.

Example 6. Supposed fz : Rn+1 → R is a generic k-parameter family, z ∈ Dk, of the form

fz(x1, ...xn+1) = q(x1, ..., xn) + dz(xn+1)

where q is a non-degenerate quadratic form in the variables x1, ..., xn and dz : R→ R is a generic k-parameterfamily as in Example 5. Suppose µz and µ′z are smooth k-parameter families of metrics on Rn and on R

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respectively and let Rn+1 be given the direct sum metric. Let ηz be the corresponding gradient of fz. Asin Example 5 it is easy to analyze the behavior of dz: the suspension principle in §5 below shows that theintersections of the stable and unstable manifolds of the critical poitns of the fz are the same as those forthe critical poitns of dz.

1.3.1 “Nice” families of gradient-like vector fields.

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