The rational strong Novikov conjecture, the groupof volume preserving diffeomorphisms, and

Hilbert-Hadamard spaces

Jianchao WuPenn State University

ICCM, June 13, 2019

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Layers of data on a (Riemannian) manifold

Riemannian metric structure, e.g., curvatures

smooth structure

homeomorphism type

homotopy type

Rigidity phenomena: sometimes lower-level data determines orobstructs higher-level data.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Motivation 1: positive scalar curvatures

Question

Given a smooth manifold, can we assign a Riemannian metric witha prescribed scalar metric function?

[Kazdan-Werner] ⇒ the most interesting case is realizing(everywhere) positive scalar curvature. It has global obstructions.Two methods:

• Minimal hypersurface method [Schoen-Yau, 79 & 17].→ • Index theory of Dirac operators for spin manifolds

[Atiyah-Singer, Lichnerowicz]. Main idea:

/D2︸︷︷︸ = ∇∗∇︸ ︷︷ ︸ + 1

4 · k︸︷︷︸(Dirac operator)2 Laplacian(≥ 0) scalar curvature

Thus, if k > 0 everywhere, /D is invertible.⇒ Ind( /D) := dim(ker /D)− dim(coker /D) = 0.

[Atiyah-Singer]: Ind( /D) = A genus, a topological invariant.

Hence, if A genus 6= 0, then no positive scalar curvature!Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Higher index = more subtle information

The conjecture of Gromov-Lawson

An aspherical manifold does not support any positive scalar metric.

Aspherical: universal cover M is contractible.

Difficulty: the method is often not sharp enough for non-simplyconnected manifolds, e.g., tori. need higher index theory!

Write M = M/Γ where Γ = π1(M). The Dirac /D on M

can be lifted to the Dirac /D on M

Baby case: first assume G is finite.

⇒ Ker( /D) and Ker( /D∗) are finite-dim’l representations of G .

Higher index: IndG ( /D) ∈ R[G ] = { formal sums ofrepresentations of Γ }, the representation ring of Γ.For a general Γ = π1(M), there is a higher index IndG ( /D) inK0(C ∗

r (Γ)), whereC∗

r (Γ) is the reduced group C∗-algebra of Γ, andKi (−) denotes operator K -theory, for i = 0, 1.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Computation of higher index IndΓ( /D) ∈ Ki(C∗r (Γ))

Difficulty: Ki (C∗r (Γ)) is often hard to compute.

Good news #1: In practice, we only care about whether thehigher index is zero or not.Good news #2: There is an assembly map

µΓ : KΓi (EΓ)→ Ki (C

∗r (Γ))

EΓ is the universal space of Γ.KΓi (EΓ) is its K -homology group, for i = 0, 1.

IndG(-) factors through µΓ:

/DIndΓ //

f∗��

K∗(C∗r (Γ))

KΓ∗ (EΓ)

µΓ

99

K -homology (of spaces) is relatively easy to compute: e.g.,X = X1 ∪ X2 equivariantly Mayer-Vietoris exact sequence

. . .→ KΓ0 (X1 ∩ X2)→ KΓ

0 (X1)⊕ KΓ0 (X2)→ KΓ

0 (X )→ . . .Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

The Rational Strong Novikov Conjecture

A possible algorithm for detecting nonzero higher index:

1 Consider

/DIndΓ //

f∗��

K∗(C ∗r (Γ))

KΓ∗ (EΓ)

µΓ

88

2 If µΓ is injective, then IndΓ( /D) 6= 0 ⇔ f∗( /D) 6= 0.⇒ carry out the computation on the left (easier).

Since we mostly care about the torsion-free part, we want:

The Rational Strong Novikov Conjecture

µΓ : Q⊗ KΓ∗ (EΓ)→ Q⊗ K∗(C ∗

r (Γ)) is injective.

We have:

the rational strong Novikov conj. ⇒ Gromov-Lawson conj.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Motivation 2: the (classical) Novikov conjecture

The Borel conjecture (an ultimate goal)

Among aspherical smooth manifolds, homotopy equivalence implieshomeomorphism.

The Novikov conjecture is an “infinitesimal version” of this.The Novikov conjecture

The higher signatures of smooth orientable manifolds are invariantunder oriented homotopy equivalences.

Remark: Higher signatures, a priori, depend on theRiemannian structure, but Novikov proved they arehomeomorphism invariants.

Following the work of Mischenko and Kasparov and using thesignature operator in place of the Dirac operator in the higherindex machinery, we obtain:

the rational strong Novikov conj. ⇒ the Novikov conj.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

The Rational Strong Novikov Conjecture (for a countable group Γ)

µΓ : Q⊗ KΓ∗ (EΓ)→ Q⊗ K∗(C ∗

r (Γ)) is injective.

Some milestone results:1 subgroups of GL(n,R) [Guentner-Higson-Weinberger],2 hyperbolic groups [Connes-Moscovici, Kasparov-Skandalis, Lafforgue],3 groups acting isometrically & properly on Hadamard manifolds

(simply connected complete Riemannian manifolds withnon-positive sectional curvature, e.g., Rn & hyperbolic spaces)[Kasparov],

4 a-T-menable groups (e.g., amenable groups) [Higson-Kasparov].

All these results make use of certain “non-positive curvature”property of the groups.

For example, a-T-menable groups, by definition, actsisometrically and properly on Hilbert spaces (“curvature=0”).

We ask: can we unite (3) and (4) and generalize them to aclass of metric spaces including both Hilbert spaces andHadamard manifolds?

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Main result

Common generalization of Hilbert spaces and Hadamard manifolds?

Definition

A Hilbert-Hadamard space is a complete CAT(0) metric spacewhose tangent cones embed isometrically into Hilbert spaces.

Such a space M is admissible if M =⋃

n Mn for (Mn)n anincreasing sequence of closed convex subsets isometric to(finite-dim’l) Riemannian manifolds.

Think: “Infinite dimensional analogs of Hadamard manifolds”.

Theorem (Gong-W-Yu)

The rational strong Novikov conjecture holds for groups actingisometrically & properly on an admissible Hilbert-Hadamard space

Baby example of such an action on an Hadamard manifold:SL(n,Z) y SL(n,R)/SO(n).Remark: its Baum-Connes conjecture is unknown.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Main example: volume-preserving diffeomorphism groups

Notice SL(n,R)/SO(n) ∼= P(n) := {positive definite n × nmatrices with determinant 1}.

Let N be a closed n-dimensional smooth manifold.

Let ω be a density on N (a “Lebesgue-like” measure).

the space of all L2-Riemannian metrics on (N, ω):L2(N, ω,P(n)) = {measurable and “L2-integrable” functions

ξ : N → P(n)} / “measure zero differences”,equipped with a metric given by

d(ξ, η)2 =

∫NdP(n)(ξ(x), η(x))2 dω(x) .

This is an admissible Hilbert-Hadamard space.

The actual Riemannian metrics sit densely in L2(N, ω,P(n)).

If Γ is a group of diffeomorphisms of N preserving ω, then Γacts isometrically on L2(N, ω,P(n)) by pushing forward(L2-)Riemannian metrics.

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

More on actions on L2(N , ω,P(n)): tame vs. wild

Let Γ be a group of diffeomorphisms of N preserving ω. There aretwo very different cases:

Case 1: Γ fixes a metric g on N ⇒ Γ y L2(N, ω,P(n)) fixes apoint. Then Γ < Isom(N, g), a compact Lie group ⇒ itsatisfies Novikov by [Guentner-Higson-Weinberger].

Case 2: Γ y L2(N, ω,P(n)) is metrically proper. Thishappens iff the length function

Γ 3 γ 7→(∫

Nlog2(‖Dxγ‖) dω(x)

) 12

is proper, where Dxγ : TxN → Tγ·xN is the derivative of thediffeomorphism γ at x and ‖ · ‖ is the operator norm. Thisfalls into the scope of our theorem.

Question

What about the general case?

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard

Thank you!

Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard