The Novikov conjecture for geometrically discrete groups of - PowerPoint PPT Presentation

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: Higher signatures: ⇓ metric structure, e.g., curvatures • by Definition � f ∗ ( x ) ∪ L ( M ) , [ M ] � , Rigidity • ⇓ smooth structure ← − − − − − − − − nomena ⇑ ? phe- where x ∈ H ∗ ( B Γ , Q ), Novikov Thm ⇓ homeomorphism type • ← − − − − − − − − Γ = π 1 ( M ), Novikov Conj • homotopy type ← − − − − − − − − and f : M → B Γ. 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 of 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: µ Γ : K ∗ ( B Γ) ⊗ Z Q → K ∗ ( C ∗ r (Γ)) ⊗ Z Q . � �� � � �� � K-theory of the reduced group C ∗ -algebra of Γ K -homology of the classifying space 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 . � �� � γ →∞ ∀ / ∃ x ∈ H , d ( x , γ · x ) − − − − → ∞ 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 d X ( x , m ) ≤ d R 2 ( 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 M n for ( M n ) 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 ℓ 2 -metric). https://www.math.tamu.edu/ ∼ jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms 6 / 13

More examples? Take L 2 -continuum products . Construction: L 2 -continuum products • X = metric space • ( Y , µ ) = finite measure space � metric space � � L 2 ( Y , µ, X ) := ξ : Y → X | measurable and L 2 -integrable � �� � � d X ( ξ ( y ) , η ( y )) 2 d µ ( y ). such that d ( ξ, η ) 2 := d ( ξ, const) < ∞ Y • Fact: X is admissible Hilbert-Hadamard and ( Y , µ ) is separable ⇒ L 2 ( Y , µ, X ) is admissible Hilbert-Hadamard. Main example: the space of L 2 -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 R n with a fixed volume form } ∼ = { positive definite n Ö n -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 } L 2 -completion → space of L 2 -Riem. metrics L 2 ( 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 L 2 -Riemannian metrics L 2 ( 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 ) � L 2 ( N , ω, SL ( n , R ) / SO ( n )) =: M isometrically by “pushing forward L 2 -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) (Guentner-Higson-Weinberger) = ⇒ Γ < Isom ( N , g ) = ⇒ Novikov Conj. γ →∞ “wild”: Γ � M properly (i.e., d ( ξ, γ · ξ ) − − − − → ∞ ) ⇐ ⇒ Γ is (what we call) geometrically discrete , i.e., � � � 1 / 2 γ →∞ (log � D y γ � ) 2 d ω ( y ) λ + ( γ ) := − − − → ∞ y ∈ N where D y γ : T y N → T γ · y N 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 ∗ ( B Γ) ⊗ Z Q → K ∗ ( C ∗ r Γ) ⊗ Z Q is injective, we construct a suitable Bott map and show the composition is injective: µ β → K ∗ ( C ∗ K ∗ ( B Γ) ⊗ Z Q r Γ) ⊗ Z Q → K ∗ + j ( A ⋊ r Γ) ⊗ Z Q How to choose the coefficient algebra A ? If Γ � R n isom. & properly, we may choose A = C 0 ( R n ). Or better, we choose A = C 0 ( R n , Cliff( R n )) to simplify the construction of the Bott map β . Or even better, we choose � f ∈ C 0 ( R n × [0 , ∞ ) , Cliff( R n ⊕ R ) | A = A ( R n ) = f ( R n × { 0 } ) ⊂ Cliff( R n ⊕ 0) � because A ( R n ) ∼ KK C 0 ( R ) ⇒ fixed dimension shift. Difficulty: If Γ � ∞ -dim’l Hilbert-Hadamard space M (e.g., H ), then typically C 0 ( 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 ( R n ) � � f ∈ C 0 ( R n × R ≥ 0 , Cliff( R n ⊕ R ) | f ( R n ×{ 0 } ) ⊂ Cliff( R n ⊕ 0) = . • We give an intrinsic construction of A ( H ), not using R n ր H . • Our construction works for all Hilbert-Hadamard spaces. Construction: homomorphisms inducing C 0 ( R ) ∼ KK A ( R n ) Fix x 0 ∈ R n � unbounded Clifford multiplier C x 0 on A ( R n ) C x 0 : R n × R ≥ 0 ∋ ( x , t ) �→ ( x , t ) − ( x 0 , 0) ∈ R n ⊕ R ֒ → Cliff( R n ⊕ R ) � Bott homomorphism β x 0 : C 0 ( R ) → A ( R n ), f �→ f ( C x 0 ). ⇒ A ( R n ) is generated by Obs: Stone-Weierstraß = � Cliff( R n ⊕ R ). { f ( C x 0 ) | f ∈ C 0 ( R ) , x 0 ∈ R n } , say, inside R n × R ≥ 0 Obs: For any Hilbert-Hadamard space M , we can define Clifford � multipliers on Cliff(span( T x M ) ⊕ R ) (using geodesics) . ( x , t ) ∈M× R ≥ 0 Definition: the C ∗ -algebra of a Hilbert-Hadamard space M A ( M ) := C ∗ -alg generated by all f ( C x 0 ) inside the above product. https://www.math.tamu.edu/ ∼ jwu/slides/NovikovDiff-MFO.pdf The Novikov conjecture and groups of diffeomorphisms10 / 13