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The Novikov conjecture for geometrically discrete groups of diffeomorphisms

Jianchao Wu

Texas A&M University Based on joint work with Sherry Gong and Guoliang Yu

Oberwolfach Workshop “Non-Commutative Geometry and Cyclic Homology”, July 2, 2020

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 1 / 13

The Novikov conjecture

Layers of data on a Riemannian manifold:

Rigidity phe- nomena ⇑?

• ⇓ metric structure, e.g., curvatures
• ⇓ smooth structure

by Definition

← − − − − − − − −

• ⇓ homeomorphism type

Novikov Thm

← − − − − − − − −

• homotopy type

Novikov Conj

← − − − − − − − −

Higher signatures:

f ∗(x) ∪ L(M), [M], where x ∈ H∗(BΓ, Q), Γ = π1(M), and f : M → BΓ.

The Novikov Conjecture The higher signatures of smooth orientable manifolds are invariant under oriented homotopy equivalences. NCG provides surprising and very successful approaches. Theorem (Connes ’83) The Novikov conjecture holds for the Gel’fand-Fuchs classes of a group of diffeomorphisms of a closed smooth manifold. This result is a display of the power of cyclic cohomology. We also focus on groups of diffeomorphisms, but with a different approach.

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 2 / 13

The Rational Strong Novikov Conjecture

Another NCG approach is to prove the following strenghening

• f the Novikov conjecture:

The Rational Strong Novikov Conjecture For any countable group Γ, the following higher index map (a variant of the Baum-Connes assembly) is injective: µΓ : K∗(BΓ) ⊗Z Q → K∗(C ∗

r (Γ)) ⊗Z Q.

• K-homology of the classifying space of Γ

K-theory of the reduced group C ∗-algebra of Γ “topological” “analytical” easier to calculate difficult to calculate

Bonus: The rational strong Novikov conjecture also implies the Gromov-Lawson conjecture. The Gromov-Lawson Conjecture An aspherical manifold cannot have positive scalar curvature.

• Homotopy structure

Metric structure

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 3 / 13

Some milestone results (a very incomplete list!)

The rational strong Novikov conjecture holds for Γ if... Theorem (Higson-Kasparov ’01) ... Γ is a-T-menable, i.e., has the Haagerup property, i.e., it acts isometrically and (metrically) properly on a Hilbert space H.

• ∀/∃x ∈ H, d(x, γ · x)

γ→∞

− − − − → ∞

They actually proved the Baum-Connes Conj. for these groups. Some further results involving Hilbert space geometry: Theorem (Yu ’00, Skandalis-Tu-Yu ’02) ... Γ coarsely embeds into a Hilbert space H. Vast classes of groups have been verified to satisfy this, e.g.: Theorem (Guentner-Higson-Weinberger ’05) ... Γ is a subgroup of a Lie group or GL(n, R) for an abelian ring R. Example: The isometry group Isom(M, g) of a Riem. mfd.

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 4 / 13

Some milestone results (a very incomplete list!), cont’d

The rational strong Novikov conjecture holds for Γ if... Theorem (Kasparov ’88) ... Γ acts isometrically and properly on an Hadamard manifold.

• = a simply connected complete Riemannian manifold with sectional curvature ≤ 0

= a complete CAT(0) Riemannian manifold

Definition A geodesic metric space X is CAT(0) if for any geodesic triangle and its Euclidean comparison triangle as follows: , where m and m are midpoints, we have dX(x, m) ≤ dR2(x, m). Example: Γ = SL(n, Z) SL(n, R)/SO(n).

• symmetric space of noncompact type

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 5 / 13

Main result

Common generalization of Hilbert spaces and Hadamard manifolds? Theorem (Gong-W-Yu) The rational strong Novikov conjecture holds for groups acting isometrically and properly on admissible Hilbert-Hadamard spaces. Definitions A Hilbert-Hadamard space is a complete CAT(0) metric space whose tangent cones embed isometrically into Hilbert spaces.

• = a metric generalization of tangent spaces for Riemannian manifolds

Such a space M is admissible if M =

n Mn for (Mn)n an

increasing sequence of closed convex subsets isometric to (finite-dimensional) Riemannian manifolds. Think: “Infinite dimensional analogs of Hadamard manifolds”. Examples: Hilbert spaces, Hadamard manifolds, etc. Non-example: trees. Closed under taking Cartesian products (using the ℓ2-metric).

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 6 / 13

More examples? Take L2-continuum products.

Construction: L2-continuum products

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

metric space L2(Y , µ, X) :=

• ξ : Y → X | measurable and L2-integrable
• such that d(ξ, η)2 :=
• Y

dX(ξ(y), η(y))2 dµ(y).

d(ξ,const)<∞

• Fact: X is admissible Hilbert-Hadamard and (Y , µ) is separable

⇒ L2(Y , µ, X) is admissible Hilbert-Hadamard. Main example: the space of L2-Riemannian metrics

• N = a closed smooth manifold
• a density on N

= a “Lebesgue-like measure”, or a “volume form without orientation”

• Observation: {inner products on Rn with a fixed volume form}

∼ = {positive definite nÖn-matrices with det. 1} ∼ = SL(n, R)/SO(n) ⇒ • {Riemannian metrics on N inducing ω} ∼ = {smooth sections of an SL(n, R)/SO(n)-bundle over N}

L2-completion

− − − − − − − − → space of L2-Riem. metrics L2(N, ω, SL(n, R)/SO(n)).

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 7 / 13

Application: volume-preserving diffeomorphism groups

Previous slide ⇒ the space of L2-Riemannian metrics L2(N, ω, SL(n, R)/SO(n)) is admissible Hilbert-Hadamard. Diff(N, ω) := the group of diffeomorphisms of N preserving ω

(independent of ω up to group isomorphism).

Observation: Diff(N, ω) L2(N, ω, SL(n, R)/SO(n)) =: M isometrically by “pushing forward L2-Riemannian metrics”.

• For a countable Γ < Diff(N, ω), there are 2 very different cases:

“tame”: Γ fixes a metric g on N (⇒ Γ M fixes a point) = ⇒ Γ < Isom(N, g)

(Guentner-Higson-Weinberger)

= ⇒ Novikov Conj. “wild”: Γ M properly (i.e., d(ξ, γ · ξ)

γ→∞

− − − − → ∞)

⇐ ⇒ Γ is (what we call) geometrically discrete, i.e., λ+(γ) :=

y∈N

(log Dyγ)2 dω(y) 1/2 γ→∞ − − − → ∞

where Dyγ : TyN → Tγ·yN is the derivative and · is the operator norm. (Gong-W-Yu)

= ⇒ Novikov Conj. Question: How to deal with the general case?

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 8 / 13

Proof ingredient 1: basic strategy

Strategy To prove the assembly map µ: K∗(BΓ) ⊗Z Q → K∗(C ∗

r Γ) ⊗Z Q is

injective, we construct a suitable Bott map and show the composition is injective: K∗(BΓ) ⊗Z Q

µ

→ K∗(C ∗

r Γ) ⊗Z Q β

→ K∗+j(A ⋊r Γ) ⊗Z Q How to choose the coefficient algebra A? If Γ Rn isom. & properly, we may choose A = C0(Rn).

Or better, we choose A = C0(Rn, Cliff(Rn)) to simplify the construction of the Bott map β. Or even better, we choose A = A(Rn) =

• f ∈ C0(Rn × [0, ∞), Cliff(Rn ⊕ R) |

f (Rn × {0}) ⊂ Cliff(Rn ⊕ 0)

• because A(Rn) ∼KK C0(R)

⇒ fixed dimension shift.

Difficulty: If Γ ∞-dim’l Hilbert-Hadamard space M (e.g., H), then typically C0(M) = 0, thus useless.

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 9 / 13

Proof ingredient 2: C ∗-alg from a Hilbert-Hadamard space

• Higson-Kasparov constructed A(H) as a direct limit of A(Rn)

=

• f ∈ C0(Rn ×R≥0, Cliff(Rn ⊕R) | f (Rn ×{0}) ⊂ Cliff(Rn ⊕0)
• .
• We give an intrinsic construction of A(H), not using Rn ր H.
• Our construction works for all Hilbert-Hadamard spaces.

Construction: homomorphisms inducing C0(R) ∼KK A(Rn) Fix x0 ∈ Rn unbounded Clifford multiplier Cx0 on A(Rn) Cx0 : Rn ×R≥0 ∋ (x, t) → (x, t)−(x0, 0) ∈ Rn ⊕R ֒ → Cliff(Rn ⊕R) Bott homomorphism βx0 : C0(R) → A(Rn), f → f (Cx0). Obs: Stone-Weierstraß = ⇒ A(Rn) is generated by {f (Cx0) | f ∈ C0(R), x0 ∈ Rn}, say, inside

• Rn×R≥0

Cliff(Rn ⊕ R). Obs: For any Hilbert-Hadamard space M, we can define Clifford multipliers on

• (x,t)∈M×R≥0

Cliff(span(TxM) ⊕ R)

(using geodesics).

Definition: the C ∗-algebra of a Hilbert-Hadamard space M A(M) := C ∗-alg generated by all f (Cx0) inside the above product.

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms10 / 13

Proof ingredient 3: deforming the Γ-action

• Previous slide: A(M) := C ∗-alg generated by all f (Cx0).
• When M is admissible, we can show the Bott homomorphisms

βx0 : C0(R) → A(M), f → f (Cx0) induce injections on K-theory.

• Not clear if they induce bijections =

⇒ We cannot use the standard Dirac-dual-Dirac method (cannot apply the five lemma): K∗(BΓ) ⊗Z Q ∼ = K Γ

∗ (EΓ) ⊗Z Q µ

• β

injective?

• K∗(C ∗

r Γ) ⊗Z Q β

• K Γ

∗+1(EΓ, A(M)) ⊗Z Q µ

K∗+1(A(M) ⋊r Γ) ⊗Z Q

• Instead, we developed a new deformation technique:

1 Replace M by M[0,1] := L2([0, 1], M) (∞-dim’l Hilbert-Hadamard!) 2 Γ

α

M isom. & properly Γ

α

M[0,1] isom. & properly

3 ∃ homotopy of actions αt : ΓM[0,1] by “freezing” α on [t, 1]

= ⇒ α1 = α and α0 = id.

4

K Γ,α

∗

(EΓ, A(M[0,1])) ∼ = K Γ,id

∗

(EΓ, A(M[0,1])) ∼ = K∗(BΓ, A(M[0,1])).

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms11 / 13

Proof ingredient 4: KK-theory with real coefficients

• If Γ is torsion-free, we can finish the proof:

K∗+1(BΓ) ⊗Z Q ∼ =

b/c K¨ unneth formula β injective

• K Γ

∗+1(EΓ) ⊗Z Q µ

• β
• K∗+1(C ∗

r Γ) ⊗Z β

• K∗(BΓ, A(M[0,1])) ∼

=K Γ

∗ (EΓ, A(M[0,1])) ⊗Z Q µ ∼ =

K∗(A(M[0,1]) ⋊r Γ) b/c our deformation technique b/c

• Γ A(M) properly;
• n M[0,1] “K-trivializes” the action.
• EΓ = EΓ.

Recall: M[0,1] := L2([0, 1], M) (∞-dim’l Hilbert-Hadamard!)

• If Γ has torsion, we use KK-theory with real coefficients

(Antonini-Azzali-Skandalis), particularly, the fact the natural map K Γ

R,∗(EΓ, A) → K Γ R,∗(EΓ, A)

is injective. Then we obtain a similar but larger diagram.

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms12 / 13

Questions and Problems

1 Compute the K-theory of A(M) for (admissible)

Hilbert-Hadamard spaces.

If A(M) ∼KK H C0(R), we get integral strong Novikov. Potential applications to index theory of ∞-dimensional spaces.

2 How important is admissibility?

Is every separable Hilbert-Hadamard space admissible? Can we do K-theoretic computations without admissibility?

3 Coarse embeddings to and from Hilbert-Hadamard spaces.

Note SL(n, R)/SO(n) ֒ →coarse H, but the induced map L2(N, ω, SL(n, R)/SO(n)) → L2(N, ω, H) is not a coarse embedding! Can we embed into nice Banach spaces?

4 Prove Novikov Conj. for all groups of diffeomorphisms.

Thank you!

https://www.math.tamu.edu/∼jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms13 / 13