THE NOVIKOV CONJECTURE, GROUPS OF

DIFFEOMORPHISMS, AND INFINITE DIMENSIONAL

NONPOSITIVELY CURVED SPACES

JIANCHAO WU

BASED ON JOINT WORK WITH SHERRY GONG AND GUOLIANG YU

JUNE 24, 2020

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Motivations: rigidity phenomena of manifolds.

Layers of data on a Riemannian manifold M .

• Riemannian metric structure, e.g., curvatures• smooth structure• homeomorphism type• homotopy type

Rigidity phenomena: sometimes lower-level data determine or obstruct higher-level data.

The Borel conjecture. For aspherical manifolds, homotopy equivalence implieshomeomorphism.

• homeomorphism type• homotopy type

The Novikov conjecture. The higher signatures of smooth orientable manifoldsare invariant under oriented homotopy equivalences.

• smooth structure• homeomorphism type• homotopy type

The Gromov-Lawson conjecture. An aspherical manifold does not support anypositive scalar metric.

• Riemannian metric structure, e.g., curvatures• homotopy type

An NCG approach via higher index

Both the Novikov conjecture and the Gromov-Lawson conjecture are implied by:

The Rational Strong Novikov Conjecture. The rational Baum-Connes as-sembly map

µ : K∗(BΓ)⊗Z Q→ K∗(C∗rΓ)⊗Z Q

is injective.

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Some milestone results (a very incomplete list!)

The rational strong Novikov conjecture holds for a discrete group Γ if...

• [Higson-Kasparov] ...Γ acts isometrically and properly on a Hilbert spaceH.⇔ Γ is a-T-menable⇔ Γ has the Haagerup propertye.g., Γ is amenable or Γ is a free group.

More results in this direction:– [Yu] ...Γ coarsely embeds into a Hilbert space H.– ⇒ [Guentner-Higson-Weinberger] ...Γ is a subgroup of a Lie group orGL(n,R) for an abelian ring R.

Example: The isometry group Isom(M, g) of a Riemannian manifold

Question: What about groups of “nonlinear nature”, e.g., groups ofdiffeomorphisms?[Connes] The Novikov conjecture holds for the Gel’fand-Fuchs classesof groups of diffeomorphisms.

• [Kasparov] ...Γ acts isometrically and properly on a Hadamard manifold(a simply connected complete Riemannian manifold with non-positive sec-tional curvature

Definition. A geodesic metric space is CAT(0) if for any geodesic triangle

we have

Examples:– Γ = Zn y Rn– Γ = Fuchsian y H2 (hyperbolic plane)– Γ = SL(n,Z) y SL(n,R)/SO(n)

A key aspect in these results: extracting nice geometric properties of thegroups!

Question: What kind of geometric properties can we use for groups of diffeomor-phisms?

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The space of Riemannian metrics

• N = a closed smooth manifold• ω = a volume form (or a density) on N

Observation. {positive definite n×n-matrices with determinant 1} ∼= SL(n,R)/SO(n)

⇒{Riemannian metrics onN inducing ω} ∼= {smooth sections of an SL(n,R)/SO(n)-bundle over N}

Observation. Any Γ < Diff(N,ω) acts on this bundle smoothly and, in the fiberdirection, isometrically.

Isometric vs. proper. Consider the following situations for this action:

• ∃ An invariant section of the Γ-action = a fixed Riemannian metric g⇒ Γ < Isom(M, g)⇒ [Guentner-Higson-Weinberger] rational strong Novikov!

• In the other extreme, we may hope to tackle the case when the action is“proper” in some sense.

– One possible notion of properness is requiring

supy∈N

(‖Dyγ‖)γ→∞−→ ∞

where we fix an arbitrary Riemannian metric and apply the inducedoperator norm on the derivative Dy : TyN → Tγ·yN .

⇒ we actually have coarse embeddings Γ→ SL(n,R)/SO(n)→ H⇒ [Yu] rational strong Novikov!

Hope: Bridge the gap between these two extreme cases!

We made progress in this direction.

Idea: Study the space of L2-Riemannian metrics L2(N,ω, SL(n,R)/SO(n)) as an“∞-dimensional symmetric nonpositively curved space”.

Construction: L2-continuum product.

• X = metric space• (Y, µ) = finite measure space

metric space L2(Y, µ,X) :={ξ : Y → X : measurable and L2-integrable

}with the metric defined by d(ξ, η)2 :=

Key example: (N,ω) as before {Riemannian metrics on N inducing ω}

∼= {smooth sections of an SL(n,R)/SO(n)-bundle over N}L2-completion

the space of L2-Riemannian metrics L2(N,ω, SL(n,R)/SO(n)).

Observation. Any Γ < Diff(N,ω) acts on L2(N,ω, SL(n,R)/SO(n)) isometri-cally.

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Main results

Goal: To prove the rational strong Novikov conjecture for Γ that acts isometri-cally and properly on an “∞-dimensional nonpositively curved space”.

Theorem 1. The rational strong Novikov conjecture holds for groups acting iso-metrically and properly on an admissible Hilbert-Hadamard space.

Definition. A Hilbert-Hadamard space is a complete CAT(0) geodesic metricspace whose tangent cones embed isometrically into Hilbert spaces.

Definition. A Hilbert-Hadamard space X is admissible if ∃ X1 ⊂ X2 ⊂ . . . ⊂ Xsuch that

(1) each Xi is closed, convex and isometric to a Riemannian manifold, and

(2) X =⋃Xi.

Examples:

CAT(0) Hilbert-Hadamard admissibleHadamard manifolds X X X

Hilbert spaces X X Xtrees X Ö Ö

What about the space of L2-Riemannian metrics?

Remark: If both X and Y are (admissible) Hilbert-Hadamard spaces, then so isX × Y with the `2-metric.

Example. Any Γ < Diff(N,ω) acts on the admissible Hilbert-Hadamard spaceL2(N,ω, SL(n,R)/SO(n)) isometrically.

Question. When is this action proper?

Answer: It is proper iff Γ is geometrically discrete, i.e.,

λ+(γ) :=

∫y∈N

(log ‖Dyγ‖)2 dω(y)γ→∞−→ ∞

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The strategy

To prove the rational Baum-Connes assembly map

µ : K∗(BΓ)⊗Z Q→ K∗(C∗rΓ)⊗Z Q

is injective, we construct a suitable Bott map and show the composition is injective:

µ : K∗(BΓ)⊗Z Q→ K∗(C∗rΓ)⊗Z Q β→ K∗(A(X) o Γ)⊗Z Q

Main ingredients:

• The construction of a C∗-algebra A(X) from a Hilbert-Hadamard space X(extending and streamlining the construction of Higson-Kasparov).

• A new deformation technique that “trivializes” the action of Γ on A(X).

• (For groups with torsion,) use the KK-theory with real coefficients ofAntonini-Azzali-Skandalis (in particular the TAF space construction).

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Questions and Problems

Problem 1. Compute the K-theory of A(X).

Problem 2. How important is admissibility?

Problem 3. Coarse embeddings to and from Hilbert-Hadamard spaces.

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