THE NOVIKOV CONJECTURE, GROUPS OF DIFFEOMORPHISMS, …jwu/slides/NovikovDiffeo-NYC.pdf · Both the...
Transcript of THE NOVIKOV CONJECTURE, GROUPS OF DIFFEOMORPHISMS, …jwu/slides/NovikovDiffeo-NYC.pdf · Both the...
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
AVAILABLE AT
HTTPS://WWW.MATH.TAMU.EDU/~JWU/SLIDES/NOVIKOVDIFFEO-NYC.PDF
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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.
AVAILABLE AT HTTPS://WWW.MATH.TAMU.EDU/~JWU/SLIDES/NOVIKOVDIFFEO-NYC.PDF5
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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