The rational strong Novikov conjecture, the group of volume preserving diffeomorphisms, and Hilbert-Hadamard spaces Jianchao Wu Penn 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 or obstructs 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 with a 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: D 2 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. D ) = � [Atiyah-Singer]: Ind( / A genus, a topological invariant. Hence, if � A genus � = 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-simply connected manifolds, e.g., tori. � need higher index theory ! Write M = � M / Γ where Γ = π 1 ( 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 : Ind G ( / D ) ∈ R [ G ] = { formal sums of representations of Γ } , the representation ring of Γ. For a general Γ = π 1 ( M ), there is a higher index Ind G ( / D ) in K 0 ( C ∗ r (Γ)), where r (Γ) is the reduced group C ∗ -algebra of Γ, and C ∗ K i ( − ) denotes operator K-theory , for i = 0 , 1. Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard
� � D ) ∈ K i ( C ∗ Computation of higher index Ind Γ ( / r (Γ)) Difficulty: K i ( C ∗ r (Γ)) is often hard to compute. Good news #1: In practice, we only care about whether the higher index is zero or not. Good news #2: There is an assembly map µ Γ : K Γ i ( E Γ) → K i ( C ∗ r (Γ)) E Γ is the universal space of Γ. K Γ i ( E Γ) is its K-homology group, for i = 0 , 1. Ind G (-) factors through µ Γ : Ind Γ � / D K ∗ ( C ∗ r (Γ)) � f ∗ µ Γ K Γ ∗ ( E Γ) K -homology (of spaces) is relatively easy to compute: e.g., X = X 1 ∪ X 2 equivariantly � Mayer-Vietoris exact sequence . . . → K Γ 0 ( X 1 ∩ X 2 ) → K Γ 0 ( X 1 ) ⊕ K Γ 0 ( X 2 ) → K Γ 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 Ind Γ � / K ∗ ( C ∗ r (Γ)) D � f ∗ µ Γ K Γ ∗ ( E Γ) 2 If µ Γ is injective, then Ind Γ ( / D ) � = 0 ⇔ f ∗ ( / D ) � = 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 µ Γ : Q ⊗ K Γ ∗ ( E Γ) → 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 implies homeomorphism. The Novikov conjecture is an “infinitesimal version” of this. The Novikov conjecture The higher signatures of smooth orientable manifolds are invariant under oriented homotopy equivalences. Remark: Higher signatures, a priori, depend on the Riemannian structure, but Novikov proved they are homeomorphism invariants. Following the work of Mischenko and Kasparov and using the signature operator in place of the Dirac operator in the higher index 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) µ Γ : Q ⊗ K Γ ∗ ( E Γ) → Q ⊗ K ∗ ( C ∗ r (Γ)) is injective. Some milestone results: subgroups of GL ( n , R ) [Guentner-Higson-Weinberger] , 1 hyperbolic groups [Connes-Moscovici, Kasparov-Skandalis, Lafforgue] , 2 groups acting isometrically & properly on Hadamard manifolds 3 (simply connected complete Riemannian manifolds with non-positive sectional curvature, e.g., R n & hyperbolic spaces) [Kasparov] , a-T-menable groups (e.g., amenable groups) [Higson-Kasparov] . 4 All these results make use of certain “non-positive curvature” property of the groups. For example, a-T-menable groups, by definition, acts isometrically and properly on Hilbert spaces (“curvature=0”). We ask: can we unite (3) and (4) and generalize them to a class of metric spaces including both Hilbert spaces and Hadamard 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 space whose tangent cones embed isometrically into Hilbert spaces. Such a space M is admissible if M = � n M n for ( M n ) n an increasing 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 acting isometrically & properly on an admissible Hilbert-Hadamard space Baby example of such an action on an Hadamard manifold: SL ( n , Z ) � 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 × n matrices 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 L 2 -Riemannian metrics on ( N , ω ): L 2 ( N , ω, P ( n )) = { measurable and “ L 2 -integrable” functions ξ : N → P ( n ) } / “measure zero differences”, equipped with a metric given by � d ( ξ, η ) 2 = d P ( n ) ( ξ ( x ) , η ( x )) 2 d ω ( x ) . N This is an admissible Hilbert-Hadamard space. The actual Riemannian metrics sit densely in L 2 ( N , ω, P ( n )). If Γ is a group of diffeomorphisms of N preserving ω , then Γ acts isometrically on L 2 ( N , ω, P ( n )) by pushing forward ( L 2 -)Riemannian metrics. Jianchao Wu The Novikov conjecture, diffeomorphisms, & Hilbert-Hadamard
More on actions on L 2 ( N , ω, P ( n )): tame vs. wild Let Γ be a group of diffeomorphisms of N preserving ω . There are two very different cases: Case 1: Γ fixes a metric g on N ⇒ Γ � L 2 ( N , ω, P ( n )) fixes a point. Then Γ < Isom ( N , g ), a compact Lie group ⇒ it satisfies Novikov by [Guentner-Higson-Weinberger]. Case 2: Γ � L 2 ( N , ω, P ( n )) is metrically proper. This happens iff the length function �� � 1 2 log 2 ( � D x γ � ) d ω ( x ) Γ ∋ γ �→ N is proper, where D x γ : T x N → T γ · x N is the derivative of the diffeomorphism γ at x and � · � is the operator norm. This falls 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
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