Higher invariants in noncommutative geometry
aa r X i v : . [ m a t h . K T ] M a y HIGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY
ZHIZHANG XIE AND GUOLIANG YU
Dedicated to Alain Connes with great admiration
Abstract.
We give a survey on higher invariants in noncommutative geometry and theirapplications to differential geometry and topology. Introduction
Geometry and topology of a smooth manifold is often governed by natural differential op-erators on the manifold. When a smooth manifold M is closed (compact without boundary),a basic invariant of these differential operators is their Fredholm index. Roughly speakingthe Fredholm index measures the size of the solution space for an infinite dimensional linearsystem associated to the operator D . More precisely, the Fredholm index of D by the for-mula: index( D ) = dim(kernel( D )) − dim(kernel( D ∗ )) . The beauty of the Fredholm index isits invariance under small perturbations and homotopy equivalence. The Fredholm index isan obstruction to invertibility of the operator. The Fredholm index of such an operator D is computed by the well-known Atiyah-Singer index theorem [AS]. The Atiyah-Singer indextheorem has important applications to geometry, topology, and mathematical physics.Alain Connes’ powerful noncommutative geometry provides the framework for a muchmore refined index theory, called higher index theory [BC, BCH, C, CM, K]. Higher in-dex theory is a far-reaching generalization of classic Fredholm index theory by taking intoconsideration of the symmetries given by the fundamental group. Let D be an elliptic dif-ferential operator on a closed manifold M of dimension n . If f M is the universal cover of M ,and e D is the lift of D onto f M , then we can define a higher index of e D in K n ( C ∗ r ( π M )),where π M is the fundamental group of M and K n ( C ∗ r ( π M )) is the K -theory of the re-duced group C ∗ -algebra C ∗ r ( π M ). This higher index is an obstruction to the invertibilityof e D and is invariant under homotopy. Higher index theory plays a fundamental role inthe studies of problems in geometry and topology such as the Novikov conjecture on homo-topy invariance of higher signatures and the Gromov-Lawson conjecture on nonexistence ofRiemmanian metrics with positive scalar curvature. Higher indices are often referred to asprimary invariants due to its homotopy invariance property. The Baum-Connes conjectureprovides an algorithm for computing the higher index [BC, BCH] while the strong Novikovconjecture predicts when the higher index vanishes [K]. When a closed manifold M carries The first author is partially supported by NSF 1500823, NSF 1800737.The second author is partially supported by NSF 1700021, NSF 1564398. a Riemannian metric with positive scalar curvature, by the Lichnerowicz formula, the Diracoperator e D on f M is invertible and hence its higher index vanishes. If M is aspherical, i.e. itsuniversal cover is contractible, then the strong Novikov conjecture predicts that the higherindex of the Dirac operator is non-zero and hence implies the Gromov-Lawson conjecturestating that any closed aspherical manifold cannot carry a Riemannian metric with positivescalar curvature [R]. Another important application of higher index theory is the Novikovconjecture [N], a central problem in topology. Roughly speaking, the Novikov conjectureclaims that compact smooth manifolds are rigid at an infinitesimal level. More precisely, theNovikov conjecture states that the higher signatures of compact oriented smooth manifoldsare invariant under orientation preserving homotopy equivalences. Recall that a compactmanifold is called aspherical if its universal cover is contractible. In the case of asphericalmanifolds, the Novikov conjecture is an infinitesimal version of the Borel conjecture, whichstates that all compact aspherical manifolds are topologically rigid, i.e. if another compactmanifold N is homotopy equivalent to the given compact aspherical manifold M , then N is homeomorphic to M . A theorem of Novikov states that the rational Pontryagin classesare invariant under orientation preserving homeomorphisms [N1]. Thus the Novikov con-jecture for compact aspherical manifolds follows from the Borel conjecture and Novikov’stheorem, since for aspherical manfolds, the information about higher signatures is equiva-lent to that of rational Pontryagin classes. In general, the Novikov conjecture follows fromthe strong Novikov conjecture when applied to the signature operator. With the help ofnoncommutative geometry, spectacular progress has been made on the Novikov conjecture.When the higher index of an elliptic operator is trivial with a given trivialization, a sec-ondary index theoretic invariant naturally arises [HR2, Roe1]. This secondary invariant iscalled the higher rho invariant. It serves as an obstruction to locality of the inverse of aninvertible elliptic operator. For example, when a closed manifold M carries a Riemannianmetric with positive scalar curvature, the Dirac operator e D on its universal cover f M is invert-ible, hence its higher index is trivial. In this case, the positive scalar curvature metric gives aspecific trivialization of the higher index, thus naturally defines a higher rho invariant. Sucha secondary index theoretic invariant is of fundamental importance for studying the space ofpositive scalar curvature metrics of a given closed spin manifold. For instance, this secondaryinvariant is an essential ingredient for measuring the size of the moduli space (under diffeo-morphism group) of positive scalar curvature metrics on a given closed spin manifold [XY1].The following is another typical situation where higher rho invariants naturally arise. Givenan orientation-preserving homotopy equivalence between two oriented closed manifolds, thehigher index of the signature operator on the disjoint union of the two manifolds (one ofthem equipped with the opposite orientation) is trivial with a trivialization given by thehomotopy equivalence. Hence such a homotopy equivalence naturally defines a higher rhoinvariant for the signature operator [HR2, Roe1]. More generally, the notion of higher rhoinvariants can be defined for homotopy equivalences between topological manifolds [Z], andthese invariants serve as a powerful tool for detecting whether a homotopy equivalence canbe “deformed” into a homeomorphism. Furthermore, the authors proved in [WXY] that the IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 3 higher rho invariant defines a group homomorphism on the structure group of a topological manifold. As an application, one can use the higher rho invariant to measure the degree ofnonrigidity of a topological manifold.Connes’ cyclic cohomology theory provides a powerful method to compute higher rhoinvariants. It turns out that the pairing of cyclic cohomology with higher rho invariants canbe computed in terms of John Lott’s higher eta invariants. This relation can be used togive an elegant approach to the higher Atiyah-Patodi-Singer index theory for manifolds withboundary and provide a potential way to construct counter examples to the Baum-Connesconjecture.The purpose of this article is to give a friendly survey on these recent developmentsof higher invariants in noncommutative geometry and their applications to geometry andtopology.
Acknowledgment.
The authors wish to thank Alain Connes for numerous inspiring dis-cussions. 2.
Geometric C ∗ -algebras In this section, we give an overview of several C ∗ -algebras naturally arising from geometryand topology. The K-theory groups of these C ∗ -algebras serve as receptacles of our higherinvariants.Let X be a proper metric space. That is, every closed ball in X is compact. An X -moduleis a separable Hilbert space equipped with a ∗ -representation of C ( X ), the algebra of allcontinuous functions on X which vanish at infinity. An X -module is called nondegenerate ifthe ∗ -representation of C ( X ) is nondegenerate. An X -module is said to be standard if nononzero function in C ( X ) acts as a compact operator.We shall first recall the concepts of propagation, local compactness, and pseudo-locality. Definition 2.1.
Let H X be a X -module and T a bounded linear operator acting on H X .(i) The propagation of T is defined to be sup { d ( x, y ) | ( x, y ) ∈ supp ( T ) } , where supp ( T )is the complement (in X × X ) of the set of points ( x, y ) ∈ X × X for which there exist f, g ∈ C ( X ) such that gT f = 0 and f ( x ) = 0, g ( y ) = 0;(ii) T is said to be locally compact if f T and T f are compact for all f ∈ C ( X );(iii) T is said to be pseudo-local if [ T, f ] is compact for all f ∈ C ( X ).Pseudo-locality is the essential property for the concept of an abstract “differential oper-ator” in K-homology theory [A, K].The following concept was introduced by Roe in his work on higher index theory fornoncompact spaces [Roe]. Definition 2.2.
Let H X be a standard nondegenerate X -module and B ( H X ) the set of allbounded linear operators on H X . The Roe algebra of X , denoted by C ∗ ( X ), is the C ∗ -algebragenerated by all locally compact operators with finite propagations in B ( H X ).The following localization algebra was introduced by [Y]. ZHIZHANG XIE AND GUOLIANG YU
Definition 2.3.
The localization algebra C ∗ L ( X ) is the C ∗ -algebra generated by all boundedand uniformly norm-continuous functions f : [0 , ∞ ) → C ∗ ( X ) such thatpropagation of f ( t ) →
0, as t → ∞ .We define C ∗ L, ( X ) to be the kernel of the evaluation map e : C ∗ L ( X ) → C ∗ ( X ) , e ( f ) = f (0) . In particular, C ∗ L, ( X ) is an ideal of C ∗ L ( X ).The localization algebra was motivated by local index theory.Now we take symmetries into consideration. Let’s assume that a discrete group Γ actsproperly on X by isometries. Let H X be a X -module equipped with a covariant unitaryrepresentation of Γ. If we denote the representation of C ( X ) by ϕ and the representationof Γ by π , this means π ( γ )( ϕ ( f ) v ) = ϕ ( f γ )( π ( γ ) v ) , where f ∈ C ( X ), γ ∈ Γ, v ∈ H X and f γ ( x ) = f ( γ − x ). In this case, we call ( H X , Γ , ϕ ) acovariant system. Definition 2.4 ([Y3]) . A covariant system ( H X , Γ , ϕ ) is called admissible if(1) the Γ-action on X is proper and cocompact;(2) H X is a nondegenerate standard X -module;(3) for each x ∈ X , the stabilizer group Γ x acts on H X regularly in the sense that the actionis isomorphic to the action of Γ x on l (Γ x ) ⊗ H for some infinite dimensional Hilbertspace H . Here Γ x acts on l (Γ x ) by translations and acts on H trivially.We remark that for each locally compact metric space X with a proper and cocompactisometric action of Γ, there exists an admissible covariant system ( H X , Γ , ϕ ). Also, we pointout that the condition (3) above is automatically satisfied if Γ acts freely on X . If noconfusion arises, we will denote an admissible covariant system ( H X , Γ , ϕ ) by H X and call itan admissible ( X, Γ)-module.
Definition 2.5.
Let X be a locally compact metric space X with a proper and cocom-pact isometric action of Γ. If H X is an admissible ( X, Γ)-module, we denote by C [ X ] Γ the ∗ -algebra of all Γ-invariant locally compact operators with finite propagations in B ( H X ). Wedefine the equivariant Roe algebra C ∗ ( X ) Γ to be the completion of C [ X ] Γ in B ( H X ).We remark that if the Γ-action on X is cocompact, then the equivariant Roe algebra C ∗ ( X ) Γ is ∗ -isomorphic to C ∗ r (Γ) ⊗ K , where C ∗ r (Γ) is the reduced group C ∗ -algebra of Γand K is the C ∗ -algebra of all compact operators. We also point out that, up to isomorphism, C ∗ ( X ) = C ∗ ( X, H X ) does not depend on the choice of the standard nondegenerate X -module H X . The same statement holds for C ∗ L ( X ), C ∗ L, ( X ) and their Γ-equivariant versions.We can also define the maximal versions of the geometric C ∗ -algebras in this section bytaking the norm completions over all ∗ -representations of their algebraic counterparts. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 5 Higher index theory and localization
In this section, we construct the higher index of an elliptic operator. We also introduce alocal index map from the K -homology group to the K -group of the localization algebra andexplain that this local index map is an isomorphism.3.1. K-homology.
We first discuss the K -homology theory of Kasparov. Let X be a locallycompact metric space with a proper and cocompact isometric action of Γ. The K -homologygroups K Γ j ( X ), j = 0 ,
1, are generated by the following cycles modulo certain equivalencerelations (cf. [K]):(i) an even cycle for K Γ0 ( X ) is a pair ( H X , F ), where H X is an admissible ( X, Γ)-moduleand F ∈ B ( H X ) such that F is Γ-equivariant, F ∗ F − I and F F ∗ − I are locallycompact and [ F, f ] =
F f − f F is compact for all f ∈ C ( X ).(ii) an odd cycle for K Γ1 ( X ) is a pair ( H X , F ), where H X is an admissible ( X, Γ)-moduleand F is a Γ-equivariant self-adjoint operator in B ( H X ) such that F − I is locallycompact and [ F, f ] is compact for all f ∈ C ( X ).Roughly speaking, the K -homology group of X is generated by abstract elliptic operatorsover X [A, K].In the general case where the action of Γ on X is not necessarily cocompact, we define K Γ i ( X ) = lim −→ Y ⊆ X K Γ i ( Y ) , where Y runs through all closed Γ-invariant subsets of X such that Y /
Γ is compact.3.2.
K-theory and boundary maps.
In this subsection, we recall the standard construc-tion of the index maps in K -theory of C ∗ -algebras. For a short exact sequence of C ∗ -algebras0 → J → A → A / J →
0, we have a six-term exact sequence in K -theory: K ( J ) / / K ( A ) / / K ( A / J ) ∂ (cid:15) (cid:15) K ( A / J ) ∂ O O K ( A ) o o K ( J ) o o Let us recall the definition of the boundary maps ∂ i .(1) Even case. Let u be an invertible element in A / J . Let v be the inverse of u in A / J .Now suppose U, V ∈ A are lifts of u and v . We define W = (cid:18) U (cid:19) (cid:18) − V (cid:19) (cid:18) U (cid:19) (cid:18) −
11 0 (cid:19) . Notice that W is invertible and a direct computation shows that W − (cid:18) U V (cid:19) ∈ J . ZHIZHANG XIE AND GUOLIANG YU
Consider the idempotent P = W (cid:18) (cid:19) W − = (cid:18) U V + U V (1 − U V ) (2 − U V )(1 − U V ) UV (1 − U V ) (1 − V U ) (cid:19) . (3.1)We have P − (cid:18) (cid:19) ∈ J . By definition, ∂ ([ u ]) := [ P ] − (cid:20)(cid:18) (cid:19)(cid:21) ∈ K ( J ) . (2) Odd case. Let q be an idempotent in A / J and Q a lift of q in A . Then ∂ ([ q ]) := [ e πiQ ] ∈ K ( J ) . Higher index map and local index map.
In this subsection, we describe the con-structions of the higher index map [BC, BCH, K, FM] and the local index map [Y, Y1].Let ( H X , F ) be an even cycle for K Γ0 ( X ). Choose a Γ-invariant locally finite open cover { U i } of X with diameter ( U i ) < c for some fixed c >
0. Let { φ i } be a Γ-invariant continuouspartition of unity subordinate to { U i } . We define F = X i φ / i F φ / i , where the sum converges in strong operator norm topology. It is not difficult to see that( H X , F ) is equivalent to ( H X , F ) in K Γ0 ( X ). By using the fact that F has finite propagation,we see that F is a multiplier of C ∗ ( X ) Γ and, is a unitary modulo C ∗ ( X ) Γ . Consider theshort exact sequence of C ∗ -algebras0 → C ∗ ( X ) Γ → M ( C ∗ ( X ) Γ ) → M ( C ∗ ( X ) Γ ) /C ∗ ( X ) Γ → M ( C ∗ ( X ) Γ ) is the multiplier algebra of C ∗ ( X ) Γ . By the construction in Section 3.2above, F produces a class ∂ ([ F ]) ∈ K ( C ∗ ( X ) Γ ). We define the higher index of ( H X , F ) tobe ∂ ([ F ]). From now on, we denote [ F ] by Ind( H X , F ) or simply Ind( F ), if no confusionarises.To define the local index of ( H X , F ), we need to use a family of partitions of unity. Moreprecisely, for each n ∈ N , let { U n,j } be a Γ-invariant locally finite open cover of X withdiameter ( U n,j ) < /n and { φ n,j } be a Γ-invariant continuous partition of unity subordinateto { U n,j } . We define F ( t ) = X j (1 − ( t − n )) φ / n,j F φ / n,j + ( t − n ) φ / n +1 ,j F φ / n +1 ,j (3.2)for t ∈ [ n, n + 1].Then F ( t ) , ≤ t < ∞ , is a multiplier of C ∗ L ( X ) Γ and a unitary modulo C ∗ L ( X ) Γ . By theconstruction in Section 3.2 above, we define ∂ ([ F ( t )]) ∈ K ( C ∗ L ( X ) Γ ) to be the local index IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 7 of ( H X , F ). If no confusion arises, we denote this local index class by Ind L ( H X , F ) or simplyInd L ( F ).Now let ( H X , F ) be an odd cycle in K Γ1 ( X ). With the same notation from above, we set q = F +12 . Then the index class of ( H X , F ) is defined to be [ e πiq ] ∈ K ( C ∗ ( X ) Γ ). For thelocal index class of ( H X , F ), we use q ( t ) = F ( t )+12 in place of q .We have the following commutative diagram: K Γ ∗ ( X ) K ∗ ( C ∗ L ( X ) Γ ) K ∗ ( C ∗ ( X ) Γ ) , IndInd L e ∗ where e ∗ is the homomorphism induced by the evaluation map e at 0.The following result was proved in the case of simplicial complexes in [Y] and the generalcase in [QR]. Theorem 3.1.
If a discrete group Γ acts properly on a locally compact space X , then thelocal index map is an isomorphism from the K -homology group K Γ ∗ ( X ) to the K-group of thelocalization algebra K ∗ ( C ∗ L ( X ) Γ ) . The Baum-Connes assembly and a local-global principle
In this section, we formulate the Baum-Connes conjecture as a local-global principle anddiscuss its connection to the Novikov conjecture.We first recall the concept of Rips complexes.
Definition 4.1.
Let Γ be a discrete group, let F ⊆ Γ be a finite symmetric subset containingthe identity (symmetric in the sense if g ∈ F , then g − ∈ F ). The Rips complex P F (Γ) is asimplicial complex such that(i) the set of vertices is Γ;(ii) a finite subset { γ , · · · , γ n } span a simplex if and only if γ − i γ j ∈ F for all 0 ≤ i, j ≤ n .We endow the Rips complex with the simplicial metric, i.e. the maximal metric whoserestriction to a maximal simplex is the standard Euclidean metric on the simplex.The Baum-Connes conjecture [BC, BCH] can be stated as follows. Conjecture 4.2 (Baum-Connes Conjecture) . The evaluation map e induces an isomor-phism e ∗ from the K -group of the equivariant localization algebra lim −→ F K ∗ ( C ∗ L ( P F (Γ)) Γ ) tothe K -group of the equivariant Roe algebra lim −→ F K ∗ ( C ∗ ( P F (Γ)) Γ ), where the limit is takenover the directed set of all finite symmetric subset F of Γ containing the identity. ZHIZHANG XIE AND GUOLIANG YU
Note that lim −→ F K ∗ ( C ∗ ( P F (Γ)) Γ ) is isomorphic to K-group of C ∗ r (Γ), the reduced group C ∗ -algebra of Γ since the Γ action on the Rips complex is cocompact.While the K-theory of the equivariant Roe algebra is global and hard to compute, theK-theory of the localization algebra is local and completely computable. Thus the Baum-Connes conjecture is a local-global principle. If true, the conjecture provides an algorithmfor computing K -groups of equivariant Roe algebras and higher indices of elliptic operators.In particular, in this case, we see that every element in the K -group of the equivariant Roealgebra can be localized.More generally, if A is a C ∗ -algebra with an action of Γ, then we can define the equivariantRoe algebra with coefficients in A , denoted by C ∗ ( P F (Γ) , A ) Γ . The equivariant Roe algebrawith coefficients in A is ∗ -isomorphic to ( A ⋊ Γ) ⊗ K , where K is the algebra of compactoperators on a Hilbert space. We can similarly introduce an equivariant localization algebrawith coefficients to formulate the Baum-Connes conjecture with coefficients.Higson and Kasparov developed an index theory of certain differential operators on aninfinite-dimensional Hilbert space and proved the following spectacular result [HK]. Theorem 4.3.
If a discrete group Γ acts on Hilbert space properly and isomentrically, thenthe Baum-Connes conjecture with coefficients holds for Γ . Recall that an isometric action α of a group Γ on a Hilbert space H is said to be properif k α ( γ ) h k → ∞ when γ → ∞ for any h ∈ H , i.e. for any h ∈ H and any positive number R >
0, there exists a finite subset F of Γ such that k α ( γ ) h k > R if γ ∈ Γ − F . A theoremof Bekka-Cherix-Valette states that an amenable group acts properly and isometrically ona Hilbert space [BCV]. Roughly speaking, a group is amenable if there exist large finitesubsets of the group with small boundary. The concept of amenability is a large scalegeometric property and was introduced by von Neumann. We refer the readers to the book[NY] as a general reference for geometric group theory related to the Novikov conjecture.The following deep theorem is due to Lafforgue [L1]. Theorem 4.4.
The Baum-Connes conjecture with coefficients holds for hyperbolic groups.
Earlier Lafforgue developed a Banach KK-theory to attack the Baum-Connes conjecture[L]. This approach yielded the Baum-Connes conjecture for hyperbolic groups [L, MY].The Baum-Connes conjecture with coefficients actually fail for general groups. Higson-Lafforgue-Skandalis gave a counter-example in [HLS]. On the other hand, the Baum-Connesconjecture (without coefficients) is still open.5.
The Novikov conjecture
A central problem in topology is the Novikov conjecture. Roughly speaking, the Novikovconjecture claims that compact smooth manifolds are rigid at an infinitesimal level. Moreprecisely, the Novikov conjecture states that the higher signatures of compact orientedsmooth manifolds are invariant under orientation preserving homotopy equivalences. Re-call that a compact manifold is called aspherical if its universal cover is contractible. In
IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 9 the case of aspherical manifolds, the Novikov conjecture is an infinitesimal version of theBorel conjecture, which states that all compact aspherical manifolds are topologically rigid,i.e. if another compact manifold N is homotopy equivalent to the given compact asphericalmanifold M , then N is homeomorphic to M . A theorem of Novikov says that the rationalPontryagin classes are invariant under orientation preserving homeomorphisms [N1]. Thusthe Novikov conjecture for compact aspherical manifolds follows from the Borel conjectureand Novikov’s theorem, since for aspherical manfolds, the information about higher signa-tures is equivalent to that of rational Pontryagin classes. In general, the Novikov conjecturefollows from the (rational) strong Novikov conjecture.The (rational) strong Novikov conjecture can be stated as follows. Conjecture 5.1 (Strong Novikov Conjecture) . The evaluation map e induces an injection e ∗ from the K -group of the equivariant localization algebra lim −→ F K ∗ ( C ∗ L ( P F (Γ)) Γ ) to the K -group of the equivariant Roe algebra lim −→ F K ∗ ( C ∗ ( P F (Γ)) Γ ), where the limit is taken over thedirected set of all finite symmetric subset F of Γ containing the identity. The rational strongNovikov conjecture states that e ∗ is an injection after tensoring with Q .The strong Novikov conjecture predicts when the higher index of an elliptic operator isnon-zero. The strong Novikov conjecture implies the following analytic Novikov conjecture. Conjecture 5.2 (Analytic Novikov Conjecture) . The evaluation map e induces an injec-tion e ∗ from the K -group of the equivariant localization algebra K ∗ ( C ∗ L ( E Γ) Γ ) to the K -group of the equivariant Roe algebra K ∗ ( C ∗ ( E Γ) Γ ), where K ∗ ( C ∗ L ( E Γ) Γ ) is defined to belim −→ X K ∗ ( C ∗ L ( X ) Γ ) with the limit to be taken over the directed set of locally compact, Γ-equivariant, Γ-cocompact subset X of the universal space E Γ for free Γ-action, and similarly K ∗ ( C ∗ ( E Γ) Γ ) is defined to be the limit lim −→ X K ∗ ( C ∗ ( X ) Γ ). The rational analytic Novikovconjecture states that e ∗ is an injection after tensoring with Q , that is, e ∗ : K ∗ ( C ∗ L ( E Γ) Γ ) ⊗ Q → K ∗ ( C ∗ ( E Γ) Γ ) ⊗ Q is an injection.The classical Novikov conjecture follows from the rational analytic Novikov conjecture.With the help of noncommutative geometry, spectacular progress has been made on theNovikov conjecture. It has been proven that The Novikov conjecture holds when the funda-mental group of the manifold lies in one of the following classes of groups:(1) groups acting properly and isometrically on simply connected and non-positivelycurved manifolds [K],(2) hyperbolic groups [CM],(3) groups acting properly and isometrically on Hilbert spaces [HK],(4) groups acting properly and isometrically on bolic spaces [KS],(5) groups with finite asymptotic dimension [Y1], (6) groups coarsely embeddable into Hilbert spaces [Y2][H][STY],(7) groups coarsely embeddable into Banach spaces with property (H) [KY],(8) all linear groups and subgroups of all almost connected Lie groups [GHW],(9) subgroups of the mapping class groups [Ha][Ki],(10) subgroups of Out( F n ), the outer automorphism groups of the free groups [BGH],(11) groups acting properly and isometrically on (possibly infinite dimensional) admissibleHilbert-Hadamard spaces, in particular geometrically discrete subgroups of the groupof volume preserving diffeomorphisms of any smooth compact manifold [GWY].In the first three cases, an isometric action of a discrete group Γ on a metric space X is saidto be proper if for some x ∈ X , d ( x, gx ) → ∞ as g → ∞ , i.e. for any x ∈ X and any positivenumber R >
0, there exists a finite subset F of Γ such that d ( x, gx ) > R if g ∈ Γ − F .In a tour de force, Connes proved a striking theorem that the Novikov conjecture holds forhigher signatures associated to Gelfand-Fuchs classes [C1]. Connes, Gromov, and Moscoviciproved the Novikov conjecture for higher signatures associated to Lipschitz group cohomologyclasses [CGM]. Hanke-Schick and Mathai proved the Novikov conjecture for higher signaturesassociated to group cohomology classes with degrees one and two [HS][Ma].J. Rosenberg discovered an important application of the (rational) strong Novikov conjec-ture to the existence problem of Riemannian metrics with positive scalar curvature [R]. Werefer to Rosenberg’s survey [R1] for recent developments on this topic.5.1. Non-positively curved groups and hyperbolic groups.
In this subsection, wegive a survey on the work of A. Mishchenko, G. Kasparov, A. Connes and H. Moscovici, G.Kasparov and G. Skandalis on the Novikov conjecture for non-positively curved groups andGromov’s hyperbolic groups.In [M], A. Mishchenko introduced a theory of infinite dimensional Fredholm representa-tions of discrete groups to prove the following theorem.
Theorem 5.3.
The Novikov conjecture holds if the fundamental group of a manifold actsproperly, isometrically and cocompactly on a simply connected manifold with non-positivesectional curvature.
In [K], G. Kasparov developed a bivariant K-theory, called KK-theory, to prove the fol-lowing theorem.
Theorem 5.4.
The Novikov conjecture holds if the fundamental group of a manifold actsproperly and isometrically on a simply connected manifold with non-positive sectional curva-ture.
As a consequence, G. Kasparov proved the following striking theorem.
Theorem 5.5.
The Novikov conjecture holds if the fundamental group of a manifold is adiscrete subgroup of a Lie group with finitely many connected components.
The theory of hyperbolic groups was developed by Gromov [G]. Gromov’s hyperbolicgroups are generic among all finitely presented groups. A. Connes and H. Moscovici proved
IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 11 the following spectacular theorem using powerful techniques from noncommutative geometry[CM].
Theorem 5.6.
The Novikov conjecture holds if the fundamental group of a manifold is ahyperbolic group in the sense of Gromov.
The proof of Theorem 5 . Theorem 5.7.
The Novikov conjecture holds if the fundamental group of a manifold is bolic.
Bolicity is a notion of non-positive curvature. Examples of bolic groups include groupsacting properly and isometrically on simply connected manifolds with non-positive sectionalcurvature and Gromov’s hyperbolic groups.5.2.
Amenable groups, groups with finite asymptotic dimension and coarsely em-beddable groups.
In this subsection, we give a survey on the work of Higson-Kasparov onthe Novikov conjecture for amenable groups, the work of G. Yu on the Novikov conjecture forgroups with finite asymptotic dimension, and the work of G. Yu, N. Higson, Skandalus-Tu-Yu on the Novikov conjecture for groups coarsely embeddable into Hilbert spaces. Finally wediscuss the work of Kasparov-Yu on the connection of the Novikov conjecture with Banachspace geometry.As mentioned above (Theorem 4 . Theorem 5.8.
The Novikov conjecture holds if the fundamental group of a manifold actsproperly and isometrically on a Hilbert space.
Since amenable groups act properly and isometrically on a Hilbert space [BCV], the abovetheorem has the following immediate corollary.
Corollary.
The Novikov conjecture holds if the fundamental group of a manifold is amenable.
This corollary is quite striking since the geometry of amenable groups can be very com-plicated (for example, the Grigorchuk’s groups [Gr]).Next we recall a few basic concepts from geometric group theory. A non-negative function l on a countable group G is called a length function if (1) l ( g − ) = l ( g ) for all g ∈ G ; (2) l ( gh ) ≤ l ( g ) + l ( h ) for all g and h in G ; (3) l ( g ) = 0 if and only if g = e , the identityelement of G . We can associate a left-invariant length metric d l to l : d l ( g, h ) = l ( g − h ) forall g, h ∈ G . A length metric is called proper if the length function is a proper map (i.e. theinverse image of every compact set is finite in this case). It is not difficult to show that everycountable group G has a proper length metric. If l and l ′ are two proper length functions on G , then their associated length metrics are coarsely equivalent. If G is a finitely generated group and S is a finite symmetric generating set (symmetric in the sense that if an elementis in S , then its inverse is also in S ), then we can define the word length l S on G by l S ( g ) = min { n : g = s · · · s n , s i ∈ S } . If S and S ′ are two finite symmetric generating sets of G , then their associated proper lengthmetrics are quasi-isometric.The following concept is due to Gromov [G1]. Definition 5.9.
The asymptotic dimension of a proper metric space X is the smallestinteger n such that for every r >
0, there exists a uniformly bounded cover { U i } for whichthe number of U i intersecting each r ball B ( x, r ) is at most n + 1.For example, the asymptotic dimension of Z n is n and the asymptotic dimension of thefree group F n with n generators is 1. The asymptotic dimension is invariant under coarseequivalence. The Lie group GL ( n, R ) with a left invariant Riemannian metric is quasi-isometric to T ( n, R ), the subgroup of invertible upper triangular matrices. By permanenceproperties of asymptotic dimension [BD1], we know that the solvable group T ( n, R ) has finiteasymptotic dimension. As a consequence, every countable discrete subgroup of GL ( n, R )has finite asymptotic dimension (as a metric space with a proper length metric). Moregenerally one can prove that every discrete subgroup of an almost connected Lie group hasfinite asymptotic dimension (a Lie group is said to be almost connected if the number ofits connected components is finite). Gromov’s hyperbolic groups have finite asymptoticdimension [Roe2]. Mapping class groups also have finite asymptotic dimension [BBF].In [Y1], G. Yu developed a quantitative operator K-theory to prove the following theorem. Theorem 5.10.
The Novikov conjecture holds if the fundamental group of a manifold hasfinite asymptotic dimension.
The basic idea of the proof is that the finiteness of asymptotic dimension allows us todevelop an algorithm to compute K-theory in a quantitative way. This strategy has foundapplications to topological rigidity of manifolds [GTY].The following concept of Gromov makes precise of the idea of drawing a good picture ofa metric space in a Hilbert space.
Definition 5.11. (Gromov): Let X be a metric space and H be a Hilbert space. A map f : X → H is said to be a coarse embedding if there exist non-decreasing functions ρ and ρ on [0 , ∞ ) such that(1) ρ ( d ( x, y )) ≤ d H ( f ( x ) , f ( y )) ≤ ρ ( d ( x, y )) for all x, y ∈ X ;(2) lim r → + ∞ ρ ( r ) = + ∞ .Coarse embeddability of a countable group is independent of the choice of proper lengthmetrics. Examples of groups coarsely embeddable into Hilbert space include groups actingproperly and isometrically on a Hilbert space (in particular amenable groups [BCV]), groupswith Property A [Y2], countable subgroups of connected Lie groups [GHW], hyperbolic IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 13 groups [S], groups with finite asymptotic dimension, Coxeter groups [DJ], mapping classgroups [Ki, Ha], and semi-direct products of groups of the above types.The following theorem unifies the above theorems.
Theorem 5.12.
The Novikov conjecture holds if the fundamental group of a manifold iscoarsely embeddable into Hilbert space.
Roughly speaking, this theorem says if we can draw a good picture of the fundamentalgroup in a Hilbert space, then we can recognize the manifold at an infinitesimal level. Thistheorem was proved by G. Yu when the classifying space of the fundamental group has thehomotopy type of a finite CW complex [Y2] and this finiteness condition was removed byN. Higson [H], Skandalis-Tu-Yu [STY]. The original proof of the above result makes heavyuse of infinite diimensional analysis. More recently, R. Willett and G. Yu found a relativelyelementary proof within the framework of basic operator K-theory [WiY].E. Guentner, N. Higson and S. Weinberger proved the beautiful theorem that linear groupsare coarsely embeddable into Hilbert space [GHW]. Recall that a group is called linear if itis a subgroup of GL ( n, k ) for some field k . The following theorem follows as a consequence[GHW]. Theorem 5.13.
The Novikov conjecture holds if the fundamental group of a manifold is alinear group.
More recently, Bestvina-Guirardel-Horbez proved that Out( F n ), the outer automorphismgroups of the free groups, is coarsely embeddable into Hilbert space. This implies the fol-lowing theorem [BGH]. Theorem 5.14.
The Novikov conjecture holds if the fundamental group of a manifold is asubgroup of
Out( F n ) . We have the following open question.
Open Question . Is every countable subgroup of the diffeomorphism group of the circlecoarsely embeddable into Hilbert space?Let E be the smallest class of groups which include all groups coarsely embeddable intoHilbert space and is closed under direct limit. Recall that if I is a directed set and { G i } i ∈ I is a direct system of groups over I , then we can define the direct limit lim G i . We emphasizethat here the homomorphism φ ij : G i → G j for i ≤ j is not necessarily injective.The following result is a consequence of Theorem 5 . Theorem 5.16.
The Novikov conjecture holds if the fundamental group of a manifold is inthe class E . The following open question is a challenge to geometric group theorists.
Open Question . Is there any countable group not in the class E ? We mention that the Gromov monster groups are in the class E [G2, G3, AD, O].Next we shall discuss the connection of the Novikov conjecture with geometry of Banachspaces. Definition 5.18.
A Banach space X is said to have Property (H) if there exist an increasingsequence of finite dimensional subspaces { V n } of X and an increasing sequence of finitedimensional subspaces { W n } of a Hilbert space such that(1) V = ∪ n V n is dense in X ,(2) if W = ∪ n W n , S ( V ) and S ( W ) are respectively the unit spheres of V and W , thenthere exists a uniformly continuous map ψ : S ( V ) → S ( W ) such that the restrictionof ψ to S ( V n ) is a homeomorphism (or more generally a degree one map) onto S ( W n )for each n. As an example, let X be the Banach space l p ( N ) for some p ≥
1. Let V n and W n berespectively the subspaces of l p ( N ) and l ( N ) consisting of all sequences whose coordinatesare zero after the n -th terms. We define a map ψ from S ( V ) to S ( W ) by ψ ( c , · · · , c k , · · · ) = ( c | c | p/ − , · · · , c k | c k | p/ − , · · · ) .ψ is called the Mazur map. It is not difficult to verify that ψ satisfies the conditions in thedefinition of Property (H). For each p ≥
1, we can similarly prove that C p , the Banach spaceof all Schatten p -class operators on a Hilbert space, has Property (H).Kasparov and Yu proved the following. Theorem 5.19.
The Novikov conjecture holds if the fundamental group of a manifold iscoarsely embeddable into a Banach space with Property (H).
Let c be the Banach space consisting of all sequences of real numbers that are convergentto 0 with the supremum norm. Open Question . Does the Banach space c have Property (H)?A positive answer to this question would imply the Novikov conjecture since every count-able group admits a coarse embedding into c [BG].A less ambitious question is the following. Open Question . Is every countable subgroup of the diffeomorphism group of a compactsmooth manifold coarsely embeddable into C p for some p ≥ p > q ≥
2, it is also an open question to construct a bounded geometry spacewhich is coarsely embeddable into l p ( N ) but not l q ( N ). Beautiful results in [JR] and [MN]indicate that such a construction should be possible. Once such a metric space is constructed,the next natural question is to construct countable groups which coarsely contain such ametric space. These groups would be from another universe and would be different from anygroup we currently know. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 15
Gelfand-Fuchs classes, the group of volume preserving diffeomorphisms, Hilbert-Hadamard spaces.
In this subsection, we give an overview on the work of A. Connes,Connes-Gromov-Moscovici on the Novikov conjecture for Gelfand-Fuchs classes and the re-cent work of Gong-Wu-Yu on the Novikov conjecture for groups acting properly and iso-metrically on a Hilbert-Hadamard spaces and for any geometrically discrete subgroup of thegroup of volume preserving diffeomorphisms of a compact smooth manifold.A. Connes proved the following deep theorem on the Novikov conjecture [C1].
Theorem 5.22.
The Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs cohomology classes of a subgroup of the group of diffeomorphisms of a compact smoothmanifold.
The proof of this theorem uses the full power of noncommutative geometry [C].
Open Question . Does the Novikov conjecture hold for any subgroup of the group ofdiffeomorphisms of a compact smooth manifold?Motivated in part by this open question, S. Gong, J. Wu and G. Yu proved the followingtheorem [GWY].
Theorem 5.24.
The Novikov conjecture holds for groups acting properly and isometricallyon an admissible Hilbert-Hadamard space.
Roughly speaking, Hilbert-Hadamard spaces are (possibly infinite dimensional) simplyconnected spaces with non-positive curvature. We will give a precise definition a little later.We say that a Hilbert-Hadamard space M is admissible if it has a sequence of subspaces M n , whose union is dense in M , such that each M n , seen with its inherited metric from M ,is isometric to a finite-dimensional Riemannian manifold. Examples of admissible Hilbert-Hadamard spaces include all simply connected and non-positively curved Riemannian mani-fold, the Hilbert space, and certain infinite dimensional symmetric spaces. Theorem 4.3 canbe viewed as a generalization of both Theorem 2.1 and Theorem 3.1.Infinite dimensional symmetric spaces are often naturally admissible Hilbert-Hadamardspaces. One such an example is L ( N, ω,
SL( n, R ) / SO( n )) , where N is a compact smooth manifold with a given volume form ω . This infinite-dimensionalsymmetric space is defined to be the completion of the space of all smooth maps from N to X = SL( n, R ) / SO( n ) with respect to the following distance: d ( ξ, η ) = (cid:18)Z N ( d X ( ξ ( y ) , η ( y ))) dω ( y ) (cid:19) , where d X is the standard Riemannian metric on the symmetric space X and ξ and η aretwo smooth maps from N to X . This space can be considered as the space of L -metricson N with the given volume form ω and is a Hilbert-Hadamard space. With the help ofthis infinite dimensional symmetric space, the above theorem can be applied to study the Novikov conjecture for geometrically discrete subgroups of the group of volume preservingdiffeomorphisms on such a manifold.The key ingredients of the proof for Theorem 5 .
24 include a construction of a C ∗ -algebramodeled after the Hilbert-Hadamard space, a deformation technique for the isometry groupof the Hilbert-Hadamard space and its corresponding actions on K-theory, and a KK-theorywith real coefficient developed by Antonini, Azzali, and Skandalis [AAS].Let Diff( N, ω ) denote the group of volume preserving diffeomorphisms on a compact ori-entable smooth manifold N with a given volume form ω . In order to define the concept ofgeometrically discrete subgroups of Diff( N, ω ), let us fix a Riemannian metric on N with thegiven volume ω and define a length function λ on Diff( N, ω ) by: λ + ( ϕ ) = (cid:18)Z N (log( k Dϕ k )) dω (cid:19) / and λ ( ϕ ) = max (cid:8) λ + ( ϕ ) , λ + ( ϕ − ) (cid:9) for all ϕ ∈ Diff(
N, ω ), where Dϕ is the Jacobian of ϕ , and the norm k · k denotes the operatornorm, computed using the chosen Riemannian metric on N . Definition 5.25.
A subgroup Γ of Diff(
N, ω ) is said to be a geometrically discrete subgroupif λ ( γ ) → ∞ when γ → ∞ in Γ, i.e. for any R >
0, there exists a finite subset F ⊂ Γ suchthat λ ( γ ) ≥ R if γ ∈ Γ \ F .Observe that although the length function λ depends on our choice of the Riemannianmetric, the above notion of geometric discreteness does not. Also notice that if γ preservesthe Riemannian metric we chose, then λ ( γ ) = 0. This suggests that the class of geometricallydiscrete subgroups of Diff( N, ω ) does not intersect with the class of groups of isometries. Ofcourse, we already know the Novikov conjecture for any group of isometries on a compactRiemannian manifold. This, together with the following result, gives an optimistic per-spective on the open question on the Novikov conjecture for groups of volume preservingdiffeomorphisms.
Theorem 5.26.
Let N be a compact smooth manifold with a given volume form ω , andlet Diff(
N, ω ) be the group of all volume preserving diffeomorphisms of N . The Novikovconjecture holds for any geometrically discrete subgroup of Diff(
N, ω ) . Now let us give a precise definition of Hilbert-Hadamard space. We will first recall theconcept of CAT(0) spaces. Let X be a geodesic metric space. Let ∆ be a triangle in X with geodesic segments as its sides. ∆ is said to satisfy the CAT(0) inequality if there is acomparison triangle ∆ ′ in Euclidean space, with sides of the same length as the sides of ∆, such that distances between points on ∆ are less than or equal to the distances betweencorresponding points on ∆ ′ . The geodesic metric space X is said to be a CAT(0) space ifevery geodesic triangle satisfies the CAT(0) inequality. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 17
Let X be a geodesic metric space. For three distinct points x, y, z ∈ X , we define thecomparison angle e ∠ xyz to be e ∠ xyz = arccos (cid:18) d ( x, y ) + d ( y, z ) − d ( x, z ) d ( x, y ) d ( y, z ) (cid:19) . In other words, e ∠ xyz can be thought of as the angle at y of the comparison triangle ∆ xyz in the Euclidean plane.Given two nontrivial geodesic paths α and β emanating from a point p in X , meaningthat α (0) = β (0) = p , we define the angle between them, ∠ ( α, β ), to be ∠ ( α, β ) = lim s,t → e ∠ ( α ( s ) , p, β ( t )) , provided that the limit exists. For CAT(0) spaces, since the e ∠ ( α ( s ) , p, β ( t )) decreases with s and t , the angle between any two geodesic paths emanating from a point is well defined.These angles satisfy the triangle inequality.For a point p ∈ X , let Σ ′ p denote the metric space induced from the space of all geodesicsemanating from p equipped with the pseudometric of angles, that is, for geodesics α and β , we define d ( α, β ) = ∠ ( α, β ). Note, in particular, from our definition of angles, that d ( α, β ) ≤ π for any geodesics α and β .We define Σ p to be the completion of Σ ′ p with respect to the distance d . The tangent cone K p at a point p in X is then defined to be a metric space which is, as a topological space,the cone of Σ p . That is, topologically K p ≃ Σ p × [0 , ∞ ) / Σ p × { } . The metric on it is given as follows. For two points p, q ∈ K p we can express them as p = [( x, t )] and q = [( y, s )]. Then the metric is given by d ( p, q ) = p t + s − st cos( d ( x, y )) . The distance is what the distance would be if we went along geodesics in a Euclidean planewith the same angle between them as the angle between the corresponding directions in X .The following definition is inspired by [FS]. Definition 5.27. A Hilbert-Hadamard space is a complete geodesic CAT(0) metric space(i.e., an Hadamard space) all of whose tangent cones are isometrically embedded in Hilbertspaces.Every connected and simply connected
Riemannian-Hilbertian manifold with non-positivesectional curvature is a separable Hilbert-Hadamard space. In fact, a Riemannian manifoldwithout boundary is a Hilbert-Hadamard space if and only if it is complete, connected, andsimply connected, and has nonpositive sectional curvature. We remark that a CAT(0) space X is always uniquely geodesic. Recall that a subset of a geodesic metric space is called convex if it is again a geodesicmetric space when equipped with the restricted metric. We observe that a closed convexsubset of a Hilbert-Hadamard space is itself a Hilbert-Hadamard space.
Definition 5.28.
A separable Hilbert-Hadamard space M is called admissible if there isa sequence of convex subsets isometric to finite-dimensional Riemannian manifolds, whoseunion is dense in M .The notion of Hilbert-Hadamard spaces is more general than simply connected Riemannian-Hilbertian space with non-positive sectional curvature. For example, the infinite dimen-sional symmetric space L ( N, ω,
SL( n, R ) / SO( n )) is a Hilbert-Hadamard space but not aRiemannian-Hilbertian space with non-positive sectional curvature.6. Secondary invariants for Dirac operators and applications
We have been mainly concerned with the primary invariants, i.e. the higher index invari-ants, till now. Starting this section, we shall shift our focus to secondary invariants. We tryto keep the discussion relatively self-contained, which hopefully will give a better sense ofsome of the more recent development on secondary invariants.In this section, we introduce a secondary invariant for Dirac operators on manifolds withpositive scalar curvature and apply the invariant to measure the size of the moduli space ofRiemmanian metrics with positive scalar curvature on a given spin manifold.We carry out the construction in the odd dimensional case. The even dimensional case issimilar. Suppose that X is an odd dimensional complete spin manifold without boundaryand we fix a spin structure on X . Assume that there is a discrete group Γ acting on X properly and cocompactly by isometries. In addition, we assume the action of Γ preservesthe spin structure on X . A typical such example comes from a Galois cover f M of a closedspin manifold M with Γ being the group of deck transformations.Let S be the spinor bundle over M and D be the associated Dirac operator on X . Let H X = L ( X, S ) and F = D ( D + 1) − / . ( H X , F ) defines a class in K Γ1 ( X ). Note that F lies in the multiplier algebra of C ∗ ( X ) Γ ,since F can be approximated by elements of finite propagation in the multiplier algebra of C ∗ ( X ) Γ . As a result, we can directly work with F ( t ) = X j ((1 − ( t − n )) φ / n,j F φ / n,j + ( t − n ) φ / n +1 ,j F φ / n +1 ,j ) (6.1)for t ∈ [ n, n + 1]. The same index construction as before defines the index class and the localindex class of ( H X , F ). We shall denote them by Ind( D ) ∈ K ( C ∗ ( X ) Γ ) and Ind L ( D ) ∈ K ( C ∗ L ( X ) Γ ) respectively. In other words, there is no need to pass to the operator F or F ( t ) as in the general case. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 19
Now suppose in addition X is endowed with a complete Riemannian metric g whose scalarcurvature κ is positive everywhere, then the associated Dirac operator in fact naturallydefines a class in K ( C ∗ L, ( X ) Γ ). Indeed, recall that D = ∇ ∗ ∇ + κ , where ∇ : C ∞ ( X, S ) → C ∞ ( X, T ∗ X ⊗ S ) is the associated connection and ∇ ∗ is the adjointof ∇ . If κ >
0, then it follows immediately that D is invertible. So, instead of D ( D + 1) − / ,we can use F := D | D | − . Note that F +12 is a genuine projection. Define F ( t ) as in formula (6.1), and define q ( t ) := F ( t )+12 . We form the path of unitaries u ( t ) = e πiq ( t ) , ≤ t < ∞ , which defines an element in( C ∗ L ( X ) Γ ) + . Notice that u (0) = 1. So this path u ( t ) , ≤ t < ∞ , in fact lies in ( C ∗ L, ( X ) Γ ) + ,therefore defines a class in K ( C ∗ L, ( X ) Γ ).Let us now define the higher rho invariant. It was first introduced by Higson and Roe[Roe1, HR3]. Our formulation is slightly different from that of Higson and Roe. The equiv-alence of the two definitions was shown in [XY]. Definition 6.1.
The higher rho invariant ρ ( D, g ) of the pair (
D, g ) is defined to be the K -theory class [ u ( t )] ∈ K ( C ∗ L, ( X ) Γ ).The definition of higher rho invariant in the even dimensional case is similar, where oneneeds to work with the natural Z / Z -grading on the spinor bundle.Next we shall apply the higher rho invariant to estimate the size of the moduli space ofRiemannian metrics with positive scalar curvature on a given spin manifold. Let M be aclosed smooth manifold. Suppose that M carries a metric of positive scalar curvature. It iswell known that the space of all Rimennian metrics on M is contractible, hence topologicallytrivial. To the contrary, the space of all positive scalar curvature metrics on M , denoted by R + ( M ), often has very nontrivial topology. In particular, R + ( M ) is often not connectedand in fact has infinitely many connected components [BoG, LP, PS, RS]. For example, byusing the Cheeger–Gromov L -rho invariant and Lott’s delocalized eta invariant, Piazza andSchick showed that R + ( M ) has infinitely many connected components, if M is a closed spinmanifold with dim M = 4 k + 3 ≥ π ( M ) contains torsion [PS].Following Stolz [St], Weinberger and Yu introduced an abelian group P ( M ) to measurethe size of the space of positive scalar curvature metrics on a manifold M [WY]. In addition,they used the finite part of K -theory of the maximal group C ∗ -algebra C ∗ max ( π ( M )) to givea lower bound of the rank of P ( M ). A special case of their theorem states that the rank of P ( M ) is ≥
1, if M is a closed spin manifold with dim M = 2 k + 1 ≥ π ( M ) containstorsion. In particular, this implies the above theorem of Piazza and Schick.For convenience of the reader, we recall the definition of the abelian group P ( M ). Let M bean oriented smooth closed manifold with dim M ≥ π ( M ) = Γ.Assume that M carries a metric of positive scalar curvature. We denote it by g M . Let I be the closed interval [0 , M × I ) ♯ ( M × I ), where the connectedsum is performed away from the boundary of M × I . Note that π (cid:0) ( M × I ) ♯ ( M × I ) (cid:1) = Γ ∗ Γthe free product of two copies of Γ.
Definition 6.2.
We define the generalized connected sum ( M × I ) ♮ ( M × I ) to be the manifoldobtained from ( M × I ) ♯ ( M × I ) by removing the kernel of the homomorphism Γ ∗ Γ → Γthrough surgeries away from the boundary.Note that ( M × I ) ♮ ( M × I ) has four boundary components, two of them being M andthe other two being − M , where − M is the manifold M with its reversed orientation. Nowsuppose g and g are two positive scalar curvature metrics on M . We endow one boundarycomponent M with g M , and endow the two − M components with g and g . Then bythe Gromov-Lawson and Schoen-Yau surgery theorem for positive scalar curvature metrics[GL, SY], there exists a positive scalar curvature metric on ( M × I ) ♮ ( M × I ) which is aproduct metric near all boundary components. In particular, the restriction of this metricon the other boundary component M has positive scalar curvature. We denote this metricon M by g . Definition 6.3.
Two positive scalar curvature metrics g and h on M are concordant ifthere exists a positive scalar curvature metric on M × I which is a product metric near theboundary and restricts to g and h on the two boundary components respectively.One can in fact show that if g and g ′ are two positive scalar curvature metrics on M obtained from the same pair of positive scalar curvature metrics g and g by the aboveprocedure, then g and g ′ are concordant [WY]. Definition 6.4.
Fix a positive scalar curvature metric g M on M . Let P + ( M ) be the set ofall concordance classes of positive scalar metrics on M . Given [ g ] and [ g ] in P + ( M ), wedefine the sum of [ g ] and [ g ] (with respect to [ g M ]) to be [ g ] constructed from the procedureabove. Then it is not difficult to verify that P + ( M ) becomes an abelian semigroup underthis addition. We define the abelian group P ( M ) to be the Grothendieck group of P + ( M ).Recall that the group of diffeomorphisms on M , denoted by Diff( M ), acts on R + ( M ) bypulling back the metrics. The moduli space of positive scalar curvature metrics is defined tobe the quotient space R + ( M ) / Diff( M ). Similarly, Diff( M ) acts on the group P ( M ) and wedenote the coinvariant of the action by e P ( M ). That is, e P ( M ) = P ( M ) /P ( M ) where P ( M )is the subgroup of P ( M ) generated by elements of the form [ x ] − ψ ∗ [ x ] for all [ x ] ∈ P ( M )and all ψ ∈ Diff( M ). We call e P ( M ) the moduli group of positive scalar curvature metricson M . It measures the size of the moduli space of positive scalar curvature metrics on M .The following conjecture gives a lower bound for the rank of the abelian group e P ( M ). Conjecture 6.5.
Let M be a closed spin manifold with π ( M ) = Γ and dim M = 4 k − ≥ e P ( M )is ≥ N fin (Γ), where N fin (Γ) is the cardinality of the following collection of positive integers: (cid:8) d ∈ N + | ∃ γ ∈ Γ such that order( γ ) = d and γ = e (cid:9) . IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 21
In [XY1], we apply the higher rho invariants of the Dirac operator to prove the followingresult.
Theorem 6.6.
Let M be a closed spin manifold which carries a positive scalar curvaturemetric with dim M = 4 k − ≥ . If the fundamental group Γ = π ( M ) of M is stronglyfinitely embeddable into Hilbert space, then the rank of the abelian group e P ( M ) is ≥ N fin (Γ) . To prove this theorem, we need index theoretic invariants that are insensitive to the actionof the diffeomorphism group. The index theoretic techniques used in [WY], for example, donot produce such invariants. The key idea of the proof is that the higher rho invariantremains unchanged in a certain K -theory group under the action of the diffeomorphismgroup, allowing us to distinguish elements in e P ( M ).We now recall the concept of strongly finite embeddability into Hilbert space for groups[XY1]. This concept is a stronger version of the notion of finite embeddability into Hilbertspace introduced in [WY], a concept more flexible than the notion of coarse embeddability. Definition 6.7.
A countable discrete group Γ is said to be finitely embeddable into Hilbertspace H if for any finite subset F ⊆ Γ, there exist a group Γ ′ that is coarsely embeddableinto H and a map φ : F → Γ ′ such that(1) if γ, β and γβ are all in F , then φ ( γβ ) = φ ( γ ) φ ( β );(2) if γ is a finite order element in F , then order( φ ( γ )) = order( γ ).As mentioned above, Weinberger and Yu proved that Conjecture 6 . g ∈ Γ has finite order d , then we can define an idempotent in the group algebra Q Γ by: p g = 1 d ( d X k =1 g k ) . For the rest of this survey, we denote the maximal group C ∗ -algebra of Γ by C ∗ (Γ). Definition 6.8.
We define K fin0 ( C ∗ (Γ)), called the finite part of K ( C ∗ (Γ)), to be the abeliansubgroup of K ( C ∗ ( G )) generated by [ p g ] for all elements g = e in G with finite order.We remark that rationally all representations of a finite group are induced from its finitecyclic subgroups [Serre]. This explains that the finite part of K -theory, despite being con-structed using only cyclic subgroups, rationally contains all K -theory elements which can beconstructed using finite subgroups. Definition 6.9.
Let J fin0 ( C ∗ (Γ)) be the abelian subgroup of K fin0 ( C ∗ (Γ)) generated by ele-ments [ p γ ] − [ p β ] with order( γ ) = order( β ). We define the reduced finite part of K ( C ∗ (Γ))to be e K fin0 ( C ∗ (Γ)) = K fin0 ( C ∗ (Γ)) / J fin0 ( C ∗ (Γ)) . An argument in [WY] can be used to prove the following result, which plays a crucial rolein the proof of Theorem 6.2.
Proposition 6.10.
Let { γ , · · · , γ n } be a collection of nontrivial elements (i.e. γ i = e ) withdistinct finite order in Γ . We define M γ , ··· ,γ n to be the abelian subgroup of K fin ( C ∗ (Γ)) generated by { [ p γ ] , · · · , [ p γ n ] } . Let f M γ , ··· ,γ n be the image of M γ , ··· ,γ n in e K fin0 ( C ∗ (Γ)) . If Γ is finitely embeddable into Hilbert space, then(1) the abelian group f M γ , ··· ,γ n has rank n ,(2) any nonzero element in K fin0 ( C ∗ (Γ)) is not in the image of the assembly map µ : K Γ0 ( E Γ) → K ( C ∗ (Γ)) , where E Γ is the universal space for proper and free Γ -action. So one is led to the following conjecture.
Conjecture 6.11.
Let Γ be a countable discrete group. Suppose { γ , · · · , γ n } is a collectionof elements in Γ with distinct finite orders and γ i = e for all 1 ≤ i ≤ n . Then (1) the abelian group f M γ , ··· ,γ n has rank n , (2) any nonzero element in K fin0 ( C ∗ (Γ)) is not in the image of the assembly map µ : K Γ0 ( E Γ) → K ( C ∗ (Γ)) , where E Γ is the universal space for proper and free Γ-action.We are now ready to introduce the notion of strongly finitely embeddability for groups.Since we are interested in the fundamental groups of manifolds, all groups are assumed tobe finitely generated in the following discussion.Let Γ be a countable discrete group. Then any set of n automorphisms of Γ, say, ψ , · · · , ψ n ∈ Aut(Γ), induces a natural action of F n the free group of n generators onΓ. More precisely, if we denote the set of generators of F n by { s , · · · , s n } , then we have ahomomorphism F n → Aut(Γ) by s i ψ i . This homomorphism induces an action of F n onΓ. We denote by Γ ⋊ { ψ , ··· ,ψ n } F n the semi-direct product of Γ and F n with respect to thisaction. If no confusion arises, we shall write Γ ⋊ F n instead of Γ ⋊ { ψ , ··· ,ψ n } F n . Definition 6.12.
A countable discrete group Γ is said to be strongly finitely embeddableinto Hilbert space H , if Γ ⋊ { ψ , ··· ,ψ n } F n is finitely embeddable into Hilbert space H for all n ∈ N and all ψ , · · · , ψ n ∈ Aut(Γ).We remark that all coarsely embeddable groups are strongly finitely embeddable. Indeed,if a group Γ is coarsely embeddable into Hilbert space, then Γ ⋊ { ψ , ··· ,ψ n } F n is coarselyembeddable (hence finitely embeddable) into Hilbert space for all n ∈ N and all ψ , · · · , ψ n ∈ Aut(Γ).If a group Γ has a torsion free normal subgroup Γ ′ such that Γ / Γ ′ is residually finite, then Γis strongly finitely embeddable into Hilbert space. Indeed, recall that any finitely generatedgroup has only finitely many distinct subgroups of a given index. Let Γ m be the intersectionof all subgroups of Γ with index at most m . Then Γ / Γ m is a finite group. Moreover, for IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 23 given ψ , · · · , ψ n ∈ Aut(Γ), the induced action of F n on Γ descends to an action of F n onΓ / Γ m . In other words, we have a natural homomorphism φ m : Γ ⋊ F n → (Γ / Γ m ) ⋊ G m where G m is the image of F n under the homomorphism F n → Aut(Γ / Γ m ). It follows that,for any finite set F ⊆ Γ, there exists a sufficiently large m such that the map φ m : F ⊂ Γ ⋊ F n → (Γ / Γ m ) ⋊ G m satisfies(1) if γ, β and γβ are all in F , then φ ( γβ ) = φ ( γ ) φ ( β );(2) if γ is a finite order element in F , then order( φ ( γ )) = order( γ ).Notice that (Γ / Γ m ) ⋊ G m is a finite group, which is obviously coarsely embeddable intoHilbert space. This shows that Γ is strongly finitely embeddable into Hilbert space.To summarize, we see that the class of strongly finitely embeddable groups includes allresidually finite groups, virtually torsion free groups (e.g. Out ( F n )), and groups that coarselyembed into Hilbert space, where the latter contains all amenable groups and Gromov’shyperbolic groups.The notion of sofic groups is a generalization of amenable groups and residually finitegroups. It is an open question whether sofic groups are (strongly) finitely embeddable intoHilbert space. Narutaka Ozawa, Denise Osin and Thomas Delzant have independently con-structed examples of groups which are not finitely embeddable into Hilbert space. An affir-mative answer to the above question would imply that there exist non-sofic groups.By definition, strongly finite embeddability implies finite embeddability. It is an openquestion whether the converse holds: Open Question . If a group is finitely embeddable into Hilbert space, then does it followthat the group is also strongly finitely embeddable into Hilbert space?In fact, it was shown in [WY] that Gromov’s monster groups and any group of analytic dif-feomorphisms of an analytic connected manifold fixing a given point are finitely embeddableinto Hilbert space. It is still an open question whether these groups are strongly finitelyembeddable into Hilbert space.Now let us proceed to prove Theorem 6 .
6. One of main ingredients of the proof is thefollowing proposition , which, combined with a surgery technique [GL, SY] and the relativehigher index theorem [B, XY2], allows us to construct genuinely “new” positive scalar cur-vature metrics from old ones. For a finite group F , an F -manifold Y is called F -connectedif the quotient Y /F is connected. Let Z d be the cyclic group of order d . Proposition 6.14.
Given positive integers d and k , there exist Z d -connected closed spin Z d -manifolds { Y , · · · , Y n } with dim Y i = 2 k such that Note that in this case, all finite order elements in Γ ⋊ { ψ , ··· ,ψ n } F n come from Γ. Proposition 6 .
14 first appeared in [WY]. The original statement in [WY] seems to contain a minor errorwhen d is even, the version we state in this survey and its proof can be found in [XYZ]. ( a ) the Z d -equivariant indices of the Dirac operators on { Y , · · · , Y n } rationally generate KO ( Z d ) ⊗ Q , ( b ) Z d acts on Y i freely except for finitely many fixed points. Let M be a closed spin manifold with a positive scalar curvature metric g M and dim M ≥ γ ∈ Γ, one can construct a new positivescalar curvature metric h γ on M such that the relative higher index Ind Γ ( g M , h γ ) = [ p γ ] ∈ K ( C ∗ (Γ)), where p γ = d P dk =1 γ k with d = order ( γ ). The detailed construction will begiven in the next paragraphs. Here let us recall the definition of this relative higher index Ind Γ ( g M , h γ ). We endow M × R with the metric g t + ( dt ) where g t is a smooth path ofRiemannian metrics on M such that g t = g M for t ≤ h γ for t ≥ g M to h γ for 0 ≤ t ≤ M × R becomes a complete Riemannian manifold with positive scalar curvature awayfrom a compact subset. Denote by D M × R the corresponding Dirac operator on M × R withrespect to this metric. Then the higher index of D M × R is well-defined and is denoted by Ind Γ ( g M , h γ ) (cf. the discussion at the beginning of Section 7 below).Next we shall describe a construction of a new positive scalar curvature metric h γ on M associated to a nontrivial finite order element γ ∈ Γ. Let f M be the universal coverof M . For each finite order element g in G with order d . By Proposition 6 .
14, thereexist Z /d Z -connected compact smooth spin Z /d Z -manifolds { N , · · · , N n } such that thedimension of each N i is 4 k and the sum of the Z /d Z -equivariant indices of the Dirac operatorson { N , · · · , N n } is a nonzero multiple of the trivial representation of Z /d Z .Let N g,l = G × Z /d Z N l , where Z /d Z acts on N l as in Proposition 6 .
14 and Z /d Z acts on G by [ m ] h = hg m for all h ∈ G and [ m ] ∈ Z /d Z . Observe that N g,l is a G -manifold.Let { g , · · · , g r } be a collection of finite order elements such that { [ p g ] , · · · , [ p g r ] } generates an abelian subgroup of K ( C ∗ ( G )) with rank r . Let N g i = F j i l =1 N g i ,l be the disjoint union of all G -manifolds described as above. Let I be the unitinterval [0 , G -equivariant connected sum ( f M × I ) ♮N g i along afree G -orbit of each N g i ,l and away from the boundary of f M × I as follows. We first obtain a G ∗ j i -equivariant connected sum ( f M × I ) ♯N g i along a free G -orbit of each N g i ,l and away fromthe boundary of ˜ M × I , where G ∗ j i is the free product of j i copies of G . More precisely, weinductively form the G ∗ j i -equivariant connected sum ( · · · (( f M × I ) ♯N g i , ) · · · ) ♯N g i ,j i , wherethe equivariant connected sum is inductively taken along a free orbit and away from theboundary. We denote this space by ( f M × I ) ♯N g i . We then perform surgeries on ( f M × I ) ♯N g i to obtain a G -equivariant cobordism between two copies of G -manifold f M .For any positive scalar curvature metric h on M , by [RW, Theorem 2.2], the above cobor-dism gives us another positive scalar curvature metric h i on M . Now the relative higher IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 25 index theorem [B, XY] implies that the relative higher index of the Dirac operator M × R associated to the positive scalar curvature metrics of h i and g M is [ p g i ] in K ( C ∗ ( G )). Asa consequence, we know that { [ h ] , · · · , [ h r ] } generates an abelian subgroup of P ( M ) withrank r .To summarize, one can construct distinct elements in P ( M ) by surgery theory and therelative higher index theorem. Moreover, these elements are distinguished by their relativehigher indices (with respect to g M ). However, to prove Theorem 6 .
6, that is, to show thatthese concordance classes of positive scalar curvature metrics remain distinct even aftermodulo the action of diffeomorphisms, we will need to use higher rho invariants (instead ofrelative higher indices) in an essential way.
Proof of Theorem . . Consider the following short exact sequence0 → C ∗ L, ( f M ) Γ → C ∗ L ( f M ) Γ → C ∗ ( f M ) Γ → f M is the universal cover of M . It induces the following six-term long exact sequence: K ( C ∗ L, ( f M ) Γ ) / / K ( C ∗ L ( f M ) Γ ) µ M / / K ( C ∗ ( f M ) Γ ) ∂ (cid:15) (cid:15) K ( C ∗ ( f M ) Γ ) O O K ( C ∗ L ( f M ) Γ ) o o K ( C ∗ L, ( f M ) Γ ) o o Recall that we have K ( C ∗ L ( f M ) Γ ) ∼ = K Γ0 ( f M ) and K ( C ∗ ( f M ) Γ ) ∼ = K ( C ∗ (Γ)).Fix a positive scalar curvature metric g M on M . For each finite order element γ ∈ Γ, wecan construct a new positive scalar curvature metric h γ on M such that the relative higherindex Ind Γ ( g M , h γ ) = [ p γ ] ∈ K ( C ∗ (Γ)) as described as above. Let us still denote by h γ (resp. g M ) the metric on f M lifted from the metric h γ (resp. g M ) on M . Let ρ ( D, h γ ) and ρ ( D, g M ) be the higher rho invariants for the pairs ( D, h γ ) and ( D, g M ), where D is the Diracoperator on f M . Then we have ∂ ([ p γ ]) = ∂ (cid:0) Ind Γ ( g M , h γ ) (cid:1) = ρ ( D, h γ ) − ρ ( D, g M ) , (6.2)(cf. [PS1, XY]).One of the key points of the proof is to construct a certain group homomorphism on e P ( M )which can be used to distinguish elements in e P ( M ). First, we define a map ̺ : P ( M ) → K ( C ∗ L, ( f M ) Γ ) by ̺ ( h ) := ρ ( D, h ) − ρ ( D, g M )for all h ∈ P ( M ). It follows from the definition of P ( M ) and [XY, Theorem 4.1] that the map ̺ is a well-defined group homomorphism. Now recall that a diffeomorphism ψ ∈ Diff( M )induces a homomorphism ψ ∗ : K ( C ∗ L, ( f M ) Γ ) → K ( C ∗ L, ( f M ) Γ ) . Let I ( C ∗ L, ( f M ) Γ ) be the subgroup of K ( C ∗ L, ( f M ) Γ ) generated by elements of the form[ x ] − ψ ∗ [ x ] for all [ x ] ∈ K ( C ∗ L, ( f M ) Γ ) and all ψ ∈ Diff( M ). We see that ̺ descends to agroup homomorphism e ̺ : e P ( M ) → K ( C ∗ L, ( f M ) Γ ) (cid:14) I ( C ∗ L, ( f M ) Γ ) . To see this, it suffices to verify that ̺ ( h ) − ̺ ( ψ ∗ ( h )) ∈ I ( C ∗ L, ( f M ) Γ )for all [ h ] ∈ P ( M ) and ψ ∈ Diff( M ). Indeed, we have ̺ ( h ) − ̺ ( ψ ∗ ( h )) = ρ ( D, h ) − ρ ( D, g M ) − (cid:0) ρ ( D, ψ ∗ ( h )) − ρ ( D, g M ) (cid:1) = ρ ( D, h ) − ρ ( D, ψ ∗ ( h ))= ρ ( D, h ) − ψ ∗ ( ρ ( D, h )) ∈ I ( C ∗ L, ( f M ) Γ ) . We remark that it is crucial to use the higher rho invariant, instead of the relative higherindex, to construct such a group homomorphism. Let us explain the subtlety here. Notethat there is in fact a well-defined group homomorphism
Ind rel : P ( M ) → K ( C ∗ (Γ)) by Ind rel ( h ) = Ind Γ ( D ; g M , h ) . The well-definedness of
Ind rel follows from the definition of P ( M ) and the relative higher index theorem [B, XY2]. However, in general, it is not clearat all whether Ind rel descends to a group homomorphism e P ( M ) → K ( C ∗ (Γ)) / I ( C ∗ (Γ)) , where I ( C ∗ (Γ)) is the subgroup of K ( C ∗ (Γ)) generated by elements of the form [ x ] − ψ ∗ [ x ]for all [ x ] ∈ K ( C ∗ (Γ)) and all ψ ∈ Diff( M ).Now for a collection of elements { γ , · · · , γ n } with distinct finite orders, we consider theassociated collection of positive scalar curvature metrics { h γ , · · · , h γ n } as before. To provethe theorem, it suffices to show that for any collection of elements { γ , · · · , γ n } with distinctfinite orders, the elements e ̺ ( h γ ) , · · · , e ̺ ( h γ n )are linearly independent in K ( C ∗ L, ( f M ) Γ ) (cid:14) I ( C ∗ L, ( f M ) Γ ).Let us assume the contrary, that is, there exist [ x ] , · · · , [ x m ] ∈ K ( C ∗ L, ( f M ) Γ ) and ψ , · · · , ψ m ∈ Diff( M ) such that n X i =1 c i ̺ ( h γ i ) = m X j =1 (cid:0) [ x j ] − ( ψ j ) ∗ [ x j ] (cid:1) , (6.3)where c , · · · , c n ∈ Z with at least one c i = 0.We denote by W the wedge sum of m circles. The fundamental group π ( W ) is the freegroup F m of m generators { s , · · · , s m } . We denote the universal cover of W by f W . Clearly, f W is the Cayley graph of F m with respect to the generating set { s , · · · , s m , s − , · · · , s − m } .Notice that F m acts on M through the diffeomorphisms ψ , · · · , ψ m . In other words, we havea homomorphism F m → Diff( M ) by s i ψ i . We define X = M × F m f W .
IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 27
Notice that π ( X ) = Γ ⋊ { ψ , ··· ,ψ m } F m . Let us write Γ ⋊ F m for Γ ⋊ { ψ , ··· ,ψ m } F m , if no confusionarises.Let e X be the universal cover of X . We have the following short exact sequence:0 → C ∗ L, ( e X ) Γ ⋊ F m → C ∗ L ( e X ) Γ ⋊ F m → C ∗ ( e X ) Γ ⋊ F m → . Recall that K ( C ∗ L ( e X ) Γ ⋊ F m ) ∼ = K Γ ⋊ F m ( e X ) and K ( C ∗ ( e X ) Γ ⋊ F m ) ∼ = K ( C ∗ (Γ ⋊ F m )). So wehave the following six-term long exact sequence: K ( C ∗ L, ( e X ) Γ ⋊ F m ) / / K Γ ⋊ F m ( e X ) / / K ( C ∗ (Γ ⋊ F m )) ∂ (cid:15) (cid:15) K ( C ∗ (Γ ⋊ F m )) O O K Γ ⋊ F m ( e X ) o o K ( C ∗ L, ( e X ) Γ ⋊ F m ) o o (6.4)Now recall the following Pimsner-Voiculescu exact sequence [PV]: L mj =1 K ( C ∗ (Γ)) P mj =1 − ( ψ j ) ∗ / / K ( C ∗ (Γ)) i ∗ / / K ( C ∗ (Γ ⋊ F m )) (cid:15) (cid:15) K ( C ∗ (Γ ⋊ F m )) O O K ( C ∗ (Γ)) o o L mj =1 K ( C ∗ (Γ)) P mj =1 − ( ψ j ) ∗ o o where ( ψ j ) ∗ is induced by ψ j and i ∗ is induced by the inclusion map of Γ into Γ ⋊ F m .Similarly, we also have the following two Pimsner-Voiculescu type exact sequences for K -homology and the K -theory groups of C ∗ L, -algebras in the diagram (6.4) above. L mi =1 K Γ0 ( f M ) P mj =1 − ( ψ j ) ∗ / / K Γ0 ( f M ) i ∗ / / K Γ ⋊ F m ( e X ) (cid:15) (cid:15) K Γ ⋊ F m ( e X ) O O K Γ1 ( f M ) o o L mi =1 K Γ1 ( f M ) P mj =1 − ( ψ j ) ∗ o o L mi =1 K ( C ∗ L, ( f M ) Γ ) P mj =1 − ( ψ j ) ∗ / / K ( C ∗ L, ( f M ) Γ ) i ∗ / / K ( C ∗ L, ( e X ) Γ ⋊ F m ) (cid:15) (cid:15) K ( C ∗ L, ( e X ) Γ ⋊ F m ) O O K ( C ∗ L, ( f M ) Γ ) o o L mi =1 K ( C ∗ L, ( f M ) Γ ) P mj =1 − ( ψ j ) ∗ o o where again ( ψ j ) ∗ and i ∗ are defined in the obvious way. Combining these Pimsner-Voiculescu exact sequences together, we have the following com-mutative diagram: ... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) / / L mj =1 K Γ0 ( f M ) (cid:15) (cid:15) σ / / K Γ0 ( f M ) (cid:15) (cid:15) i ∗ / / K Γ ⋊ F m ( e X ) µ (cid:15) (cid:15) / / / / L mj =1 K ( C ∗ (Γ)) σ / / (cid:15) (cid:15) K ( C ∗ (Γ)) i ∗ / / (cid:15) (cid:15) K ( C ∗ (Γ ⋊ F m )) ∂ Γ ⋊ Fm (cid:15) (cid:15) / / / / L mj =1 K ( C ∗ L, ( f M ) Γ ) σ / / (cid:15) (cid:15) K ( C ∗ L, ( f M ) Γ ) i ∗ / / (cid:15) (cid:15) K ( C ∗ L, ( e X ) Γ ⋊ F m ) / / (cid:15) (cid:15) ... ... ... (6.5)where σ = P mj =1 − ( ψ j ) ∗ . Notice that all rows and columns are exact.Now on one hand, if we pass Equation (6.3) to K ( C ∗ L, ( e X ) Γ ⋊ F m ) under the map i ∗ , thenit follows immediately that n X k =1 c k · i ∗ [ ̺ ( h γ k )] = 0 in K ( C ∗ L, ( e X ) Γ ⋊ F m ) , where at least one c k = 0. On the other hand, by assumption, Γ is strongly finitely em-beddable into Hilbert space. Hence Γ ⋊ F m is finitely embeddable into Hilbert space. ByProposition 6 .
10, we have the following.(i) { [ p γ ] , · · · , [ p γ n ] } generates a rank n abelian subgroup of K fin0 ( C ∗ (Γ ⋊ F m )), since γ , · · · , γ n have distinct finite orders. In other words, n X k =1 c k [ p γ k ] = 0 ∈ K fin0 ( C ∗ (Γ ⋊ F m ))if at least one c k = 0.(ii) Every nonzero element in K fin0 ( C ∗ (Γ ⋊ F m )) is not in the image of the assembly map µ : K Γ ⋊ F m ( E (Γ ⋊ F m )) → K ( C ∗ (Γ ⋊ F m )) , where E (Γ ⋊ F m ) is the universal space for proper and free Γ ⋊ F m -action. In particular,every nonzero element in K fin0 ( C ∗ (Γ ⋊ F m )) is not in the image of the map µ : K Γ ⋊ F m ( e X ) → K ( C ∗ (Γ ⋊ F m )) IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 29 in diagram (6.5). It follows that the map ∂ Γ ⋊ F m : K fin0 ( C ∗ (Γ ⋊ F m )) → K ( C ∗ L, ( e X ) Γ ⋊ F m )is injective. In other words, ∂ Γ ⋊ F m maps a nonzero element in K fin0 ( C ∗ (Γ ⋊ F m )) to anonzero element in K ( C ∗ L, ( e X ) Γ ⋊ F m ).To summarize, we have(a) P nk =1 c k [ p γ k ] = 0 in K fin0 ( C ∗ (Γ ⋊ F m )),(b) P nk =1 c k · i ∗ [ ̺ ( h γ k )] = 0 in K ( C ∗ L, ( e X ) Γ ⋊ F m ),(c) the map ∂ Γ ⋊ F m : K fin0 ( C ∗ (Γ ⋊ F m )) → K ( C ∗ L, ( e X ) Γ ⋊ F m ) is injective,(d) and by Equation (6.2), ∂ Γ ⋊ F m (cid:16) P nk =1 c k [ p γ k ] (cid:17) = P nk =1 c k · i ∗ [ ̺ ( h γ k )].Therefore, we arrive at a contradiction. This finishes the proof. (cid:3) Higher index, higher rho and positive scalar curvature at infinity
In this section, we will first describe a construction of the higher index for the Diracoperator on a complete manifold with positive scalar curvature at infinity. This constructionis due to Gromov-Lawson in the classic Fredholm case [GL1] and its generalization to higherindex case is due to Bunke [B] (see also [Roe1, BW, Roe3]). We will then discuss a connectionbetween the higher index for the Dirac operator on a manifolds with boundary and the higherrho invariant of the Dirac operator on the boundary.Let M be a complete Riemannian spin manifold with a proper and isometric action of adiscrete group Γ. We assume that M has positive scalar curvature at infinity relative to theaction of Γ, i.e. there exists a Γ-cocompact subset Z of M and a positive number a suchthat the scalar curvature k of M is greater than or equal to a outside Z . Let D be the Diracoperator M .We need some preparations in order to define the higher index. The following useful lemmais due to Roe [Roe3]. Lemma 7.1.
With the notation as above, suppose that f ∈ S ( R ) has its Fourier transform ˆ f supported in ( − r, r ) . Let φ ∈ C ( M ) have support disjoint from a r -neighborhood of Z .We have || f ( D ) φ || ≤ || φ || sup {| f ( λ ) | : | λ | ≥ a } . Proof.
Let us first deal with the case where f is an even function. In the case, the Fouriertransform formula gives us f ( D ) = Z r − r ˆ f ( t ) cos ( tD ) dt. Let us define U = { x ∈ M : d ( x, Z ) > r } and U ′ = { x ∈ M : d ( x, Z ) > r } . Consider the unbounded symmetric operator D with domain C ∞ c ( U ). This operator isbounded below by a I and has a Friedrichs extension on the Hilbert space L ( U, S ), where S is the spinor bundle. We denote this extension by E . Clearly, E is bounded below by thesame lower bound a I .A standard finite propagation speed argument shows that if s is smooth and compactlysupported on U ′ , then cos ( tD ) s = cos ( t √ E ) s for − r ≤ t ≤ r. Since the spectrum of √ E is bounded below by a , we have || f ( √ E ) || ≤ sup {| f ( λ ) | : | λ | ≥ a } . This implies the following inequality: || f ( D ) φ || ≤ || φ || sup {| f ( λ ) | : | λ | ≥ a } . If f is an odd function, we have || f ( D ) φ || ≤ || ¯ φ || || f ( D ) φ || . In this case, the function f is even, belongs to S ( R ), and has Fourier transform supportedin ( − r, r ). Hence we have the following inequality: || f ( D ) φ || ≤ || φ || sup {| f ( λ ) | : | λ | ≥ a } . It follows that || f ( D ) φ || ≤ || φ || sup {| f ( λ ) | : | λ | ≥ a } . The general case follows from the above two special cases by writing f as a sum of evenand odd functions. (cid:3) With the help of the above lemma, we can prove the following result.
Lemma 7.2.
For any f ∈ C c ( − a, a ) , we have f ( D ) ∈ lim R →∞ C ∗ ( N R ( Z )) Γ , where N R ( Z ) isthe R -neighborhood of Z and lim R →∞ C ∗ ( N R ( Z )) Γ is the C ∗ -algebra limit of the equivariantRoe algebras.Proof. For any ǫ >
0, there exists a smooth function g such that its Fourier transform iscompactly supported, and sup {| g ( λ ) − f ( λ ) | : λ ∈ R } < ǫ. It follows that | g ( λ ) | < ǫ for | λ | > a. Let r be a positive number such that Supp(ˆ g ) ⊆ ( − r, r )and let ψ : M → [0 ,
1] be a continuous Γ-invariant function equal to 1 on a 4 r -neighborhoodof Z and vanishing outside a 5 r -neighborhood of Z . We write f ( D ) = ψg ( D ) ψ + (1 − ψ ) g ( D ) ψ + g ( D )(1 − ψ ) + ( f ( D ) − g ( D )) . Note that the first term is a Γ-equivariant and locally compact operator with finite propa-gation supported near Z , the second and third terms have norm bounded by 2 ǫ by Lemma7 .
1. This implies the desired result. (cid:3) Without loss of generality, we can assume N R ( Z ) is Γ-invariant. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 31
We remark lim R →∞ C ∗ ( N R ( Z )) Γ is isomorphic to the reduced group C ∗ -algebra C ∗ r (Γ).A normalizing function χ : R → [ − ,
1] is, by definition, a continuous odd function thatgoes to ± x → ∞ . Now choose a normalizing function χ such that χ − − a, a ) and let F = χ ( D ) . By Lemma 7.2, the same construction from Section 3 defines a higher index
Ind ( D ) ∈ K ∗ ( C ∗ r (Γ)) . The following question is wide open.
Open Question . Let M be a complete spin manifold with a proper and isometric actionof a discrete group Γ. Let D be the Dirac operator on M . Assume that M has positivescalar curvature at infinity relative to the action of Γ. Is Ind ( D ) an element in the image ofthe Baum-Connes assembly map?Let N be a spin manifold with boundary, where the boundary ∂N is endowed with apositive scalar curvature metric. We will explain that the K -theoretic “boundary” of thehigher index class of the Dirac operator on N is equal to the higher rho invariant of theDirac operator on ∂N . More generally, let M be an m -dimensional complete spin manifoldwith boundary ∂M such that(i) the metric on M has product structure near ∂M and its restriction on ∂M , denoted by h , has positive scalar curvature;(ii) there is a proper and cocompact isometric action of a discrete group Γ on M ;(iii) the action of Γ preserves the spin structure of M .We denote the associated Dirac operator on M by D M and the associated Dirac operatoron ∂M by D ∂M . With the positive scalar curvature metric h on the boundary ∂M , wecan define the higher index class Ind ( D M ) of D M in K ∗ ( C ∗ r (Γ)) as follows. We can attacha cylinder ∂M × [0 , ∞ ) to the boundary of M to form a complete Riemannian manifold(without boundary) ¯ M , where the Riemannian metric on M is naturally extended to ¯ M such that Riemannian metric on the cylinder is a product. The action of Γ on M naturallyextends to an action on ¯ M . By construction, ¯ M has positive scalar curvature at infinityrelative to the action of M . We can therefore define the higher index Ind ( D M ) of D M to bethe higher index of the Dirac operator on ¯ M .Notice that the short exact sequence of C ∗ -algebras0 → C ∗ L, ( M ) Γ → C ∗ L ( M ) Γ → C ∗ ( M ) Γ → K -theory: · · · → K i ( C ∗ L ( M ) Γ ) → K i ( C ∗ ( M ) Γ ) ∂ i −→ K i − ( C ∗ L, ( M ) Γ ) → K i − ( C ∗ L ( M ) Γ ) → · · · . Also, by functoriality, we have a natural homomorphism ι ∗ : K m − ( C ∗ L, ( ∂M ) Γ ) → K m − ( C ∗ L, ( M ) Γ )induced by the inclusion map ι : ∂M ֒ → M . With the above notation, one has the followingtheorem. Theorem 7.4. ∂ m ( Ind ( D M )) = ι ∗ ( ρ ( D ∂M , h )) in K m − ( C ∗ L, ( M ) Γ ) . This theorem is due to Piazza and Schick [PS1] when the dimension of M is even and to Xieand Yu [XY] in the general case. As an immediate application, one sees that nonvanishingof the higher rho invariant is an obstruction to extension of the positive scalar curvaturemetric from the boundary to the whole manifold. Moreover, the higher rho invariant can beused to distinguish whether or not two positive scalar curvature metrics are connected by apath of positive scalar curvature metrics.8. Secondary invariants of the signature operators and topologicalnon-rigidity
In this section, we introduce the higher rho invariants for a pair of closed manifoldswhich are homotopic equivalent to each other. Roughly speaking, we consider the relativesignature operator associated to this pair of manifolds. This relative signature operator hastrivial higher index with a natural trivilialization given by the homotopy equivalence. Thistrivialization allows us to define a higher rho invariant, which can be used to detect whethera homotopy equivalence can be deformed into a homeomorphism.We shall focus on the case of smooth manifolds. General topological manifolds can behandled in a similar way with the help of Lipschitz structures [Su].Let M and N be two closed oriented smooth manifolds of dimension n . We will only discussthe odd dimensional case; the even dimensional case is completely similar. We denote thede Rham complex of differential forms on M byΩ ( M ) d −−→ Ω ( M ) d −−→ · · · d −−→ Ω n ( M ) , whose L -completion is Ω L ( M ) d −−→ Ω L ( M ) d −−→ · · · d −−→ Ω nL ( M ) . We shall write d M if we need to specify d is the differential associated to the de Rhamcomplex of M . Similarly, we haveΩ L ( N ) d N −−−→ Ω L ( N ) d N −−−→ · · · d n −−→ Ω nL ( N )for the manifold N .Let T = ∗ M : Ω kL ( M ) → Ω n − kL ( M ) be the Hodge star operator on M , which is defined by h T α, β i = Z M α ∧ β, where β is the complex conjugation of β . The Hodge star operator T satisfies the followingproperties:(1) T ∗ α = ( − k ( n − k ) T α for any α ∈ Ω kL ( M );(2) T dα + ( − k d ∗ T α = 0 for any smooth β ∈ Ω k ( M );(3) T α = ( − nk + k α for any α ∈ Ω kL ( M ); IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 33 where T ∗ is the adjoint of T , and d ∗ is the adjoint of d . More generally, a bounded operator T satisfying conditions (1) and (2) is said to be a duality operator of the chain complex(Ω ∗ L ( M ) , d ) if in addition, it satisfies the condition(3) ′ T induces a chain homotopy equivalence from the dual complex of (Ω ∗ L ( M ) , d ) to thecomplex (Ω ∗ L ( M ) , d ), where the dual complex is defined to beΩ nL ( M ) d ∗ −−→ Ω n − L ( M ) d ∗ −−→ · · · d ∗ −−→ Ω L ( M ) . In this case, we call (Ω ∗ L ( M ) , d ) together with the duality operator T a (unbounded) Hilbert-Poincar´e complex.Define S = i k ( k − m T , where m = ( n − /
2. It follows from properties (1) and (3) abovethat S is a selfadjoint involution. Definition 8.1.
The signature operator D of M is defined to be i ( d + d ∗ ) S acting on evendegree differential forms.All the above discussion generalizes to the universal covering f M of M . We denote thecorresponding π ( M )-equivariant signature operator of f M by e D .In the standard K -theoretic construction of the index of e D (cf. Section 3), let us choosethe normalizing function χ ( t ) = π arctan( t ). In this case, we have Ind ( e D ) = e πi χ ( e D )+12 = ( e D − i )( e D + i ) − . Let B = d + d ∗ . The above formula implies the following index formula: Ind ( e D ) = ( B − S )( B + S ) − ∈ K ( C ∗ r ( π ( M ))) . (8.1)The above index formula in fact holds for general Hilbert-Poincar´e complexes, that is,chain complexes with general duality operators. We shall not get into the technical detailsregarding the notion of Hilbert-Poincar´e complexes, but instead refer the reader to [HR] fordetails. A key feature of the notion of Hilbert-Poincar´e complexes is that it allows us touse a much larger class of duality operators besides the Hodge star operators. In the caseof general Hilbert-Poincar´e complexes, the well-definedness of the above index formula isjustified by the following lemma [HR, Lemma 3.5]. Lemma 8.2. B + S and B − S are invertible.Proof. Consider the mapping cone complex associated to the chain map S : (Ω ∗ L ( M ) , − d ∗ ) → (Ω ∗ L ( M ) , d )with the differential b = (cid:18) d S d ∗ (cid:19) . Since S is an isomorphism on the homology, the mapping cone complex is acyclic. Thereforethe operator b + b ∗ is invertible. Recall that S is self-adjoint. Hence we have b + b ∗ = (cid:18) d + d ∗ SS d + d ∗ (cid:19) . Note that (cid:18) d + d ∗ SS d + d ∗ (cid:19) (cid:18) vv (cid:19) = (cid:18) ( d + d ∗ + S ) v ( d + d ∗ + S ) v (cid:19) . This implies that B + S is invertible. We can similarly show that B − S is invertible. (cid:3) Suppose f : M → N is an orientation preserving homotopy equivalence between M and N . It is known that Ind ( e D M ) = Ind ( e D N ) in K ( C ∗ r (Γ)), where Γ = π ( M ) = π ( N ), cf.[K1][KM]. Intuitively speaking, one can use the homotopy equivalence f together with thesignature operators on M and N to produce an invertible operator D f on M ∪ ( − N ) suchthat the index of D f coincides with the index of the signature operator on M ∪ ( − N ), whichis Ind ( e D M ) − Ind ( e D N ), cf. [HiS]. Here − N is the manifold N with the reversed orientationand M ∪ ( − N ) stands for the disjoint union of the two. In particular, the invertibility of D f is the reason that Ind ( e D M ) = Ind ( e D N ), by giving a specific trivialization of the indexclass of D M ∪ ( − N ) . Thus the homotopy equivalence f naturally defines a higher rho invariant.In the following, we shall take a different but simpler approach to construct the higher rhoinvariant of f . Although this process does not produce an invertible operator D f , but itdoes provide a trivialization at the K-theory level. Our choice of such an approach is mainlyits simplicity, which will hopefully convey the key ideas with more clarity.We denote the induced pullback map on differential forms by f ∗ : Ω ∗ ( N ) → Ω ∗ ( M ). Ingeneral, f ∗ does not extend to a bounded linear map between the spaces of L forms Ω ∗ L ( N )and Ω ∗ L ( M ). In order to fix this issue, we need the following construction due to Hilsum andSkandalis [HiS]. First, suppose ϕ : X → Y is a submersion between two closed manifolds.It is easy to see that ϕ ∗ does extend to a bounded linear operator from Ω ∗ L ( Y ) to Ω ∗ L ( X ).Now let ι : N → R k be an embedding. Suppose U is a tubular neighborhood of N in R k and π : U → N is the associated projection. Without loss of generality, we assume ι ( N ) + B k ⊂ U , where B k is the unit ball of R k . Let p : M × B k → N be the submersiondefined by p ( x, t ) = π ( f ( x ) + t ). Furthermore, let ω be a volume form on B k whose integralis 1. Then the formula α → Z B k p ∗ ( α ) ∧ ω defines a morphism of chain complexes A : Ω ∗ ( N ) → Ω ∗ ( M ), where R B k denotes fiberwiseintegration along B k . It is easy to see that A extends to a bounded linear operator fromΩ ∗ L ( N ) to Ω ∗ L ( M ). We shall still denote this extension by A : Ω ∗ L ( N ) → Ω ∗ L ( M ).Now a routine calculation shows that A is a homotopy equivalence between the two com-plexes (Ω L ( M ) , d M ) and (Ω L ( N ) , d N ) such that AT A ∗ is chain homotopy equivalent to T ′ , IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 35 where T ′ is the Hodge star operator on N . It follows that the operator S = (cid:18) ATT A ∗ (cid:19) together with the chain complex (Ω ∗ L ( M ) ⊕ Ω ∗ L ( N ) , d M ⊕ d N ) gives rise to an (unbounded)Hilbert-Poincar´e complex.We have the following lemma due to Higson and Roe [HR]. Lemma 8.3.
If we write B = (cid:18) B M B N (cid:19) = (cid:18) d M + d ∗ M d N + d ∗ N (cid:19) , then the element ( B − S )( B + S ) − is equal to Ind ( e D M ) − Ind ( e D N ) in K ( C ∗ r (Γ)) .Proof. Note that T ′ and AT A ∗ induce the same map on homology. It follows that the path (cid:18) T
00 ( s − T ′ − sAT A ∗ (cid:19) is an operator homotopy connecting the duality operator T ⊕ − T ′ to the duality operator T ⊕ − AT A ∗ . The path (cid:18) cos ( s ) T sin ( s ) T A ∗ sin ( s ) AT − cos ( s ) AT A ∗ (cid:19) is an operator homotopy connecting the duality operator T ⊕ − AT A ∗ to the duality operator (cid:18) ATT A ∗ (cid:19) , where s ∈ [0 , π/ . Now the lemma follows from the explicit index formula inline (8.1). (cid:3)
For each t ∈ [0 , π ], the following operator S t = (cid:18) e it ATe − it T A ∗ (cid:19) defines a duality operator for the chain complex (Ω ∗ L ( M ) ⊕ Ω ∗ L ( N ) , d M ⊕ d N ). It is notdifficult to verify that ( B − S )( B + S t ) − defines a continuous path of invertible elements in ( C ∗ r (Γ) ⊗ K ) + . Note that S π = − S ,thus ( B − S )( B + S π ) − = 1. Therefore, the path ( B − S )( B + S t ) − gives a specifictrivialization of the index class Ind ( e D M ) − Ind ( e D N ). This trivialization in turn induces ahigher rho invariant as follows. Let { v ( t ) } ≤ t ≤ be the path of invertible elements connecting( B − S )( B + S ) − to (cid:0) B − ( T T ′ ) (cid:1)(cid:0) B + ( T T ′ ) (cid:1) − . We define ρ ( t ) = ( B − S )( B + S (1 − t ) π ) − , for 0 ≤ t ≤ ,v ( t ) for 1 ≤ t ≤ , e πi ( χ ( t e D M )+1) e πi ( χ ( t e D − N )+1) ! for 2 ≤ t < ∞ . Definition 8.4.
We define the higher rho invariant of a given homotopy equivalence f : M → N to be the above element [ ρ ] in K ( C ∗ L, ( e N ) Γ ). Here we have used f to map elements in C ∗ ( f M ) Γ to C ∗ ( e N ) Γ .The fact that ρ is an element in the matrix algebra of ( C ∗ L, ( e N ) Γ ) + follows from a stan-dard finite propagation speed argument. The even dimensional higher rho invariant can bedefined in a similar way. Zenobi generalized the concept of higher rho invariant to homotopyequivalences between closed topological manifolds with the help of Lipschitz structures [Z].Given a closed oriented manifold N , the higher rho invariant in fact defines a map fromthe structure set of N to K n ( C ∗ L, ( e N ) Γ ), where n = dim N . On the other hand, when N is a topological manifold, the structure set of N carries a natural abelian group structure. It waslong standing open problem whether the higher rho inviariant map is a group homomorphismfrom the structure group of N to K ( C ∗ L, ( ˜ N ) Γ ). This was answered in positive in completegenerality by Weinberger-Xie-Yu [WXY]. In the following we shall briefly discuss some ofthe key ideas of their proof and also some applications to topology.Let X be a closed oriented connected topological manifold of dimension n . The structuregroup S ( X ) is the abelian group of equivalence classes of all pairs ( f, M ) such that M isa closed oriented manifold and f : M → X is an orientation-preserving homotopy equiva-lence. Recall that the abelian group structure on S ( X ) is originally described through theSiebenmann periodicity map, which is an injection from S ( X ) to S ∂ ( X × D ), where D isthe 4-dimensional Euclidean unit ball and S ∂ ( X × D ) is the rel ∂ version of structure setof X × D . The set S ∂ ( X × D ) carries a natural abelian group structure by stacking, andinduces an abelian group structure on S ( X ) by Nicas’ correction map to the Siebenmannperiodicity map [Ni]. Both S ( X ) and S ∂ ( X × D ) carry a higher rho invariant map. It isnot difficult to verify that the higher rho invariant map on S ∂ ( X × D ) is additive, i.e. ahomomorphism between abelian groups. One possible approach to show the additivity ofthe higher rho invariant map on S ( X ) is to prove the compatibility of higher rho invariantmaps on S ( X ) and S ∂ ( X × D ). However, there are some essential analytical difficultiesto directly prove such a compatibility, due to the subtleties of the Siebenmann periodicitymap . A main novelty of Weinberger-Xie-Yu’s approach [WXY] is to give a new descriptionof the topological structure group in terms of smooth manifolds with boundary. This new A geometric construction of the Siebenmann periodicity map was given by Cappell and Weinberger [CW]
IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 37 description uses more objects and an equivalence relation broader than h -cobordism, whichallows us to replace topological manifolds in the usual definition of S ( X ) by smooth man-ifolds with boundary. Such a description leads to a transparent group structure, which isgiven by disjoint union. The main body of Weinberger-Xie-Yu’s work [WXY] is devoted toproving that the new description coincides with the classical description of the topologicalstructure group; and to developing the theory of higher rho invariants in this new setting,in which higher rho invariants are easily seen to be additive. As a consequence, the higherrho invariant maps on S ( X ) and S ∂ ( X × D ) are indeed compatible. Theorem 8.5 ([WXY, Theorem 4.40]) . The higher rho invariant map is a group homomor-phism from S ( X ) to K n ( C ∗ L, ( e X ) Γ ) . As mentioned above, the above theorem solves the long standing open problem whetherthe higher rho inviariant map defines a group homomorphism on the topological structuregroup. As an application, Weinberger-Xie-Yu applied the above theorem to prove that thestructure groups of certain manifolds are infinitely generated [WXY].
Theorem 8.6.
Let M be a closed oriented topological manifold of dimension n ≥ , and Γ be its fundamental group. Suppose the rational strong Novikov conjecture holds for Γ . If ⊕ k ∈ Z H n +1+4 k (Γ , C ) is infinitely generated, then the topological structure group of S ( M ) isinfinitely generated. We refer to the article [WXY] for examples of groups satisfying the conditions in the abovetheorem.9.
Non-rigidity of topological manifolds and reduced structure groups
The structure group measures the degree of non-rigidity and the reduced structure groupestimates the size of non-rigidity modulo self-homotopy equivalences. In this section, weapply the higher rho invariants of signature operators to give a lower bound of the freerank of reduced structure groups of closed oriented topological manifolds. Our key tool isthe additivity property of higher rho invariants from the previous section. There are infact two different versions of reduced structure groups, e S alg ( X ) and e S geom ( X ), whose precisedefinitions will be given below. The group e S alg ( X ) is functorial and fits well with the surgerylong exact sequence. On the other hand, the group e S geom ( X ) is more geometric in the sensethat it measures the size of the collection of closed manifolds homotopic equivalent but nothomeomorphic to X .Since we will be using the maximal version of various C ∗ -algebras throughout this section,we will omit the subscript “max” for notational simplicity.Let X be an n -dimensional oriented closed topological manifold. Denote the monoid oforientation-preserving self homotopy equivalences of X by Aut h ( X ). There are two differentactions of Aut h ( X ) on S ( X ), which induce two different versions of reduced structure groupsas follows. On one hand,
Aut h ( X ) acts naturally on S ( X ) by α u ( θ ) = u ∗ ( θ )for all u ∈ Aut h ( X ) and all θ ∈ S ( X ), where u ∗ is the group homomorphism from S ( X ) to S ( X ) induced by the map u [KiS]. This action α is compatible with the actions of Aut h ( X )on other terms in the topological surgery exact sequence.On the other hand, Aut h ( X ) also naturally acts on S ( X ) by compositions of homotopyequivalences, that is, β u ( θ ) = ( u ◦ f, M )for all u ∈ Aut h ( X ) and all θ = ( f, M ) ∈ S ( X ). Note that β u : S ( X ) → S ( X )only defines a bijection of sets, and is not a group homomorphism in general. Definition 9.1.
With the same notation as above, we define the following reduced structuregroups.(1) Define e S alg ( X ) to be the quotient group of S ( X ) by the subgroup generated by elementsof the form θ − α u ( θ ) for all θ ∈ S ( X ) and all u ∈ Aut h ( X ).(2) we define e S geom ( X ) to be the quotient group of S ( X ) by the subgroup generated byelements of the form θ − β u ( θ ) for all θ ∈ S ( X ) and all u ∈ Aut h ( X ).Next we recall a method of constructing elements in the structure group by the finite partof K -theory [WY, Theorem 3.4].Let M be a (4 k − π M = Γ. Suppose { g , · · · , g m } is a collection of elements in Γ with distinct finite orderssuch that g i = e for all 1 ≤ i ≤ m . Recall the topological surgery exact sequence: · · · → H k ( M, L • ) → L k (Γ) S −→ S ( M ) → H k − ( M, L • ) → · · · . For each finite subgroup H of Γ, we have the following commutative diagram: H H k ( EH, L • ) A / / (cid:15) (cid:15) L k ( H ) (cid:15) (cid:15) H G k ( E Γ , L • ) A / / L k (Γ) , where the vertical maps are induced by the inclusion homomorphism from H to Γ. For eachelement g in H with finite order d , the idempotent p g = d ( P dk =1 g k ) produces a class in L ( Q H ), where L ( Q H ) is the algebraic definition of L -groups using quadratic forms andformations with coefficients in Q . Let [ q g ] be the corresponding element in L k ( Q H ) givenby periodicity. Recall that L k ( H ) ⊗ Q ≃ L k ( Q H ) ⊗ Q . For each element g in H with finite order, we use the same notation [ q g ] to denote the elementin L k ( H ) ⊗ Q corresponding to [ q g ] ∈ L k ( Q H ) under the above isomorphism. IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 39
We also have the following commutative diagram: H Γ4 k ( E Γ , L • ) ⊗ Q A / / (cid:15) (cid:15) L k (Γ) ⊗ Q (cid:15) (cid:15) K Γ0 ( E Γ) ⊗ Q µ ∗ / / K ( C ∗ (Γ)) ⊗ Q , where the left vertical map is induced by a map at the spectra level and the right verticalmap is induced by the inclusion map: L k (Γ) → L k ( C ∗ (Γ)) ∼ = K ( C ∗ (Γ))(see [R2] for the last identification).Now if Γ is finitely embeddable into Hilbert, then the abelian subgroup of K ( C ∗ (Γ))generated by { [ p g ] , · · · , [ p g m ] } is not in the image of of the map µ ∗ : K Γ0 ( E Γ) → K ( C ∗ (Γ)) . It follows that(1) any nonzero element in the abelian subgroup of L k (Γ) ⊗ Q generated by the elements { [ q g ] , · · · , [ q g m ] } is not in the image of the rational assembly map A : H Γ4 k ( E Γ , L • ) ⊗ Q → L k (Γ) ⊗ Q ;(2) the abelian subgroup of L k (Γ) ⊗ Q generated by { [ q g ] , · · · , [ q g m ] } has rank m .By exactness of the surgery sequence, we know that the map S : L k (Γ) ⊗ Q → S ( M ) ⊗ Q , (9.1)is injective on the abelian subgroup of L k (Γ) ⊗ Q generated by { [ q g ] , · · · , [ q g n ] } .In order to prove the main result of this section, we need to apply the above argumentnot only to Γ, but also to certain semi-direct products of Γ with free groups of finitely manygenerators.Recall that N fin (Γ) is the cardinality of the following collection of positive integers: { d ∈ N + | ∃ γ ∈ Γ such that γ = e and order ( γ ) = d } . We have the following result [WXY]. At the moment, we are only able to prove the theoremfor e S alg ( M ). We will give a brief discussion to indicate the difficulties in proving the version e S geom ( M ) after the theorem. Theorem 9.2.
Let M be a closed oriented topological manifold with dimension n = 4 k − k > and π M = Γ . If Γ is strongly finitely embeddable into Hilbert space ( cf. Definition . , then the free rank of e S alg ( M ) is ≥ N fin (Γ) .Proof. A key point of the argument below is to use a semi-direct product Γ ⋊ F m to turncertain outer automorphisms of Γ into inner automorphisms of Γ ⋊ F m . Consider the higher rho invariant homomorphism from Theorem 8 . ρ : S ( M ) → K ( C ∗ L, ( f M ) Γ ) . Note that every self-homotopy equivalence ψ ∈ Aut h ( M ) induces a homomorphism e ψ ∗ : K ( C ∗ L, ( f M ) Γ ) → K ( C ∗ L, ( f M ) Γ ) . Let I ( C ∗ L, ( f M ) Γ ) be the subgroup of K ( C ∗ L, ( f M ) Γ ) generated by elements of the form[ x ] − e ψ ∗ [ x ] for all [ x ] ∈ K ( C ∗ L, ( f M ) Γ ) and all ψ ∈ Aut h ( M ). Note that, by the definition ofthe higher rho invariant, we have ρ ( α ψ ( θ )) = e ψ ∗ ( ρ ( θ )) ∈ K ( C ∗ L, ( f M ) Γ )for all θ ∈ S ( M ) and ψ ∈ Aut h ( M ). It follows that ρ descends to a group homomorphism e S alg ( M ) → K ( C ∗ L, ( f M ) Γ ) (cid:14) I ( C ∗ L, ( f M ) Γ ) . Now for a collection of elements { γ , · · · , γ ℓ } with distinct finite orders, we considerthe elements S ( p γ ) , · · · , S ( p γ ℓ ) ∈ S ( M ) as in line (9.1). To be precise, the elements S ( p γ ) , · · · , S ( p γ ℓ ) actually lie in S ( M ) ⊗ Q . Consequently, all abelian groups in the fol-lowing need to be tensored by the rationals Q . For simplicity, we shall omit ⊗ Q from ournotation, with the understanding that the abelian groups below are to be viewed as tensoredwith Q . Also, let us write ρ ( γ i ) := ρ ( S ( p γ i )) ∈ K ( C ∗ L, ( f M ) Γ ) . To prove the theorem, it suffices to show that for any collection of elements { γ , · · · , γ ℓ } withdistinct finite orders, the elements ρ ( γ ) , · · · , ρ ( γ ℓ )are linearly independent in K ( C ∗ L, ( f M ) Γ ) (cid:14) I ( C ∗ L, ( f M ) Γ ).Let us assume the contrary, that is, there exist [ x ] , · · · , , [ x m ] ∈ K ( C ∗ L, ( f M ) Γ ) and ψ , · · · , ψ m ∈ Aut h ( M ) such that ℓ X i =1 c i ρ ( γ i ) = m X j =1 (cid:0) [ x j ] − ( e ψ j ) ∗ [ x j ] (cid:1) , (9.2) Let us review how the homomorphism ψ ∗ : K ( C ∗ L, ( f M ) Γ ) → K ( C ∗ L, ( f M ) Γ ) is defined. The map ψ : M → M lifts to a map e ψ : f M → f M . However, to view e ψ as a Γ-equivariant map, we need to use twodifferent actions of Γ on f M . Let τ be a right action of Γ on f M through deck transformations. Then wedefine a new action τ ′ of Γ on f M by τ ′ g = τ ψ ∗ ( g ) , where ψ ∗ : Γ → Γ is the automorphism induced by ψ . Itis easy to see that e ψ : f M → f M is Γ-equivariant, when Γ acts on the first copy of f M by τ and the secondcopy of f M by τ ′ . Let us denote the corresponding C ∗ -algebras by C ∗ L, ( f M ) Γ τ and C ∗ L, ( f M ) Γ τ ′ . Observe that,despite the two different actions of Γ on f M , the two C ∗ -algebras C ∗ L, ( f M ) Γ τ and C ∗ L, ( f M ) Γ τ ′ are canonicallyidentical, since an operator is invariant under the action τ if and only if it is invariant under the action τ ′ . IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 41 where c , · · · , c ℓ ∈ Z with at least one c i = 0. In fact, we shall study Equation (9.2) in thegroup K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) and arrive at a contradiction, where Γ ⋊ F m is a certainsemi-direct product of Γ with the free group of m generators F m and E (Γ ⋊ F m ) is theuniversal space for free and proper Γ ⋊ F m -actions.Let us fix a map σ : M → B Γ that induces an isomorphism of their fundamental groups,where B Γ is the classifying space of Γ. Suppose ϕ : M → M is an orientation preservingself homotopy equivalence of M . Then ϕ induces an automorphism of Γ, also denoted by ϕ ∈ Aut(Γ). Now consider the semi-direct product Γ ⋊ ϕ Z and its associated classifyingspace B (Γ ⋊ ϕ Z ). Let ˆ ϕ be the element in Γ ⋊ ϕ Z that corresponds to the generator 1 ∈ Z .We write Φ : B (Γ ⋊ ϕ Z ) → B (Γ ⋊ ϕ Z )for the map induced by the automorphism Γ ⋊ ϕ Z → Γ ⋊ ϕ Z defined by a → ˆ ϕa ˆ ϕ − . Suppose ι : B Γ → B (Γ ⋊ ϕ Z ) is the map induced by the inclusion Γ ֒ → Γ ⋊ ϕ Z . Then the map ι ◦ σ ◦ ϕ : M ϕ −→ M σ −→ B Γ ι −→ B (Γ ⋊ ϕ Z )is homotopy equivalent to the mapΦ ◦ ι ◦ σ : M σ −→ B Γ ι −→ B (Γ ⋊ ϕ Z ) Φ −→ B (Γ ⋊ ϕ Z ) , since they induce the same map on fundamental groups. Let e σ : f M → E Γ be the lift of themap σ : M → B Γ. Similarly, e ϕ : f M → f M is the lift of ϕ : M → M , and e Φ : E (Γ ⋊ ϕ Z ) → E (Γ ⋊ ϕ Z ) is the lift of Φ : B (Γ ⋊ ϕ Z ) → B (Γ ⋊ ϕ Z ) . Since Φ : B (Γ ⋊ ϕ Z ) → B (Γ ⋊ ϕ Z ) is induced by an inner conjugation morphism on Γ ⋊ ϕ Z ,the map e Φ ∗ : K ( C ∗ L, ( E Γ) Γ ) → K ( C ∗ L, ( E Γ) Γ ) is the identity map. It follows that for each[ x ] ∈ K ( C ∗ L, ( f M ) Γ ), we have e ι ∗ e σ ∗ ( e ϕ ∗ [ x ]) = e Φ ∗ e ι ∗ e σ ∗ ([ x ]) = e ι ∗ e σ ∗ ([ x ])in K ( C ∗ L, ( E (Γ ⋊ ϕ Z )) Γ ⋊ ϕ Z ), where e ι ∗ e σ ∗ is the composition K ( C ∗ L, ( f M ) Γ ) e σ ∗ −→ K ( C ∗ L, ( E Γ) Γ ) e ι ∗ −→ K ( C ∗ L, ( E (Γ ⋊ ϕ Z )) Γ ⋊ ϕ Z ) . The same argument also works for an arbitrary finite number of orientation preserving selfhomotopy equivalences ψ , · · · , ψ m ∈ Aut h ( M ) simultaneously, in which case we have e ι ∗ e σ ∗ (( e ψ i ) ∗ [ x ]) = e ι ∗ e σ ∗ ([ x ]) in K ( C ∗ L, ( E (Γ ⋊ { ψ , ··· ,ψ m } F m )) Γ ⋊ { ψ , ··· ,ψm } F m ) , for all [ x ] ∈ K ( C ∗ L, ( f M ) Γ ). In other words, ( e ψ i ) ∗ [ x ] and [ x ] have the same image in K ( C ∗ L, ( E (Γ ⋊ { ψ , ··· ,ψ m } F m )) Γ ⋊ { ψ , ··· ,ψm } F m ). For notational simplicity, let us write Γ ⋊ F m Precisely speaking, ϕ only defines an outer automorphism of Γ, and one needs to make a specific choiceof a representative in Aut(Γ). In any case, any such choice will work for the proof. The C ∗ -algebra C ∗ L, ( E Γ) Γ is the inductive limit of C ∗ L, ( Y ) Γ , where Y ranges over all Γ-cocompactsubspaces of E Γ. for Γ ⋊ { ψ , ··· ,ψ m } F m . If no confusion is likely to arise, we shall still write [ x ] for its image e ι ∗ e σ ∗ ([ x ]) in K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ).If we pass Equation (9.2) to K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) under the map K ( C ∗ L, ( f M ) Γ ) e σ ∗ −→ K ( C ∗ L, ( E Γ) Γ ) e ι ∗ −→ K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) , then it follows from the above discussion that ℓ X k =1 c k ρ ( γ k ) = 0 in K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) , where at least one c k = 0. We have ∂ Γ ⋊ F m (cid:16) ℓ X k =1 c k [ p γ k ] (cid:17) = 2 · (cid:0) ℓ X k =1 c k ρ ( γ k ) (cid:1) = 0 , (9.3)where ∂ Γ ⋊ F m is the connecting map in the following long exact sequence: K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) / / K Γ ⋊ F m ( E (Γ ⋊ F m )) µ / / K ( C ∗ (Γ ⋊ F m )) ∂ Γ ⋊ Fm (cid:15) (cid:15) K ( C ∗ (Γ ⋊ F m )) O O K Γ ⋊ F m ( E (Γ ⋊ F m )) o o K ( C ∗ L, ( E (Γ ⋊ F m )) Γ ⋊ F m ) o o (9.4)Now by assumption Γ is strongly finitely embeddable into Hilbert space. Hence Γ ⋊ F m isfinitely embeddable into Hilbert space. By Proposition 6 .
10, we have the following.(i) { [ p γ ] , · · · , [ p γ ℓ ] } generates a rank n abelian subgroup of K fin0 ( C ∗ (Γ ⋊ F m )), since γ , · · · , γ n have distinct finite orders. In other words, n X k =1 c k [ p γ k ] = 0 ∈ K fin0 ( C ∗ (Γ ⋊ F m ))if at least one c k = 0.(ii) Every nonzero element in K fin0 ( C ∗ (Γ ⋊ F m )) is not in the image of the assembly map µ : K Γ ⋊ F m ( E (Γ ⋊ F m )) → K ( C ∗ (Γ ⋊ F m )) . In particular, we see that ∂ Γ ⋊ F m : K fin0 ( C ∗ (Γ ⋊ F m )) → K ( C ∗ L, ( e X ) Γ ⋊ F m ) is injective.It follows that ∂ Γ ⋊ F m (cid:16) P ℓk =1 c k [ p γ k ] (cid:17) = 0, which contradicts Equation (9.3). This finishesthe proof. (cid:3) It is tempting to use a similar argument to prove an analogue of Theorem 9 . e S geom ( M ). However, there are some subtleties. First, note that α ϕ ( θ ) + [ ϕ ] = β ϕ ( θ ) IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 43 for all θ = ( f, N ) ∈ S ( M ) and all ϕ ∈ Aut h ( M ), where [ ϕ ] = ( ϕ, M ) is the element given by ϕ : M → M in S ( M ) . It follows that ρ ( β ϕ ( θ )) = ρ ( α ϕ ( θ )) + ρ ([ ϕ ]) = ϕ ∗ ( ρ ( θ )) + ρ ([ ϕ ]) . In other words, in general, ρ ( β ϕ ( θ )) = ϕ ∗ ( ρ ( θ )), and consequently the homomorphism ρ : S ( M ) → K ( C ∗ L, ( f M ) Γ )does not descend to a homomorphism from e S geom ( M ) to K ( C ∗ L, ( f M ) Γ ) (cid:14) I ( C ∗ L, ( f M ) Γ ). Newideas are needed to take care of this issue. On the other hand, there is strong evidence whichsuggests an analogue of Theorem 9 . e S geom ( M ). For example, this has been verifiedby Weinberger and Yu for residually finite groups [WY, Theorem 3.9]. Also, Chang andWeinberger showed that the free rank of e S geom ( M ) is at least 1 when π X = Γ is not torsionfree [ChW, Theorem 1].The above discussion motivates the following conjecture. Conjecture 9.3.
Let M be a closed oriented topological manifold with dimension n = 4 k − k >
1) and π M = Γ. Then the free ranks of e S alg ( M ) and e S geom ( M ) are ≥ N fin (Γ).We conclude this section by proving the following theorem, which is an analogue of thetheorem of Chang and Weinberger cited above [ChW, Theorem 1]. Theorem 9.4.
Let X be a closed oriented topological manifold with dimension n = 4 k − k > and π X = Γ . If Γ is not torsion free, then the free rank of e S alg ( X ) is ≥ .Proof. Recall that for any non-torsion-free countable discrete group G , if γ = e is a finiteorder element of G , then [ p γ ] generates a subgroup of rank one in K ( C ∗ ( G )) and any nonzeromultiple of [ p γ ] is not in the image of the assembly map µ : K Γ0 ( EG ) → K ( C ∗ ( G )) [WY].Using this fact, the statement follows from the same proof as in Theorem 9 . (cid:3) Cyclic cohomology and higher rho invariants
Connes’ cyclic cohomology theory provides a powerful method to compute higher rhoinvariants. In this section, we give a survey of recent work on the pairing between Connes’cyclic cohomology and C ∗ -algebraic secondary invariants. In the case of higher rho invariantsgiven by invertible operators on manifolds, this pairing can be computed in terms of Lott’shigher eta invariants. We apply these results to the higher Atiyah-Patodi-Singer index theoryand discuss a potential way to construct counter examples to the Baum-Connes conjecture.We shall first discuss the zero dimensional cyclic cocycle case. Let M be a spin Riemannianmanifold with positive scalar curvature and let D be the Dirac operator on M . Let f M bethe universal cover of M and e D the lifting of D . Lott introduced the following delocalized Here “invertible” means being invertible on the universal cover of the manifold. eta invariant η h h i ( e D ) [Lo1]: η h h i ( e D ) := 2 √ π Z ∞ tr h h i ( e De − t e D ) dt, (10.1)under the condition that the conjugacy class h h i of h ∈ Γ = π M has polynomial growth.Here Γ = π M is the fundamental group of M , and the trace map tr h h i is defined as follows:tr h h i ( A ) = X g ∈h h i Z F A ( x, gx ) dx on Γ-equivariant Schwartz kernels A ∈ C ∞ ( f M × f M ), where F is a fundamental domain of f M under the action of Γ.We have the following theorem [XY3]. Theorem 10.1.
Let M be a closed odd-dimensional spin manifold equipped with a positivescalar curvature metric g . Suppose f M is the universal cover of M , ˜ g is the Riemannnianmetric on f M lifted from g , and e D is the associated Dirac operator on f M . Suppose theconjugacy class h h i of a non-identity element h ∈ π M has polynomial growth, then we have τ h ( ρ ( e D, e g )) = − η h h i ( e D ) , where ρ ( e D, e g ) is the K -theoretic higher rho invariant of e D with respect to the metric ˜ g , and τ h is a canonical determinant map associated to h h i . As an application of Theorem 10 . Theorem 10.2.
With the same notation as above, if the rational Baum-Connes conjectureholds for Γ , and the conjugacy class h h i of a non-identity element h ∈ Γ has polynomialgrowth, then the delocalized eta invariant η h h i ( e D ) is an algebraic number. Moreover, if inaddition h has infinite order, then η h h i ( e D ) vanishes. This theorem follows from the construction of the determinant map τ h and a L -Lefschetzfixed point theorem of B.-L. Wang and H. Wang [WW, Theorem 5.10]. When Γ is torsion-free and satisfies the Baum-Connes conjecture, and the conjugacy class h h i of a non-identityelement h ∈ Γ has polynomial growth, Piazza and Schick have proved the vanishing of η h h i ( e D ) by a different method [PS, Theorem 13.7].In light of this algebraicity result, we propose the following open question. Open Question . If the conjugacy class h h i of a non-identity element h ∈ Γ has polyno-mial growth, what values can the delocalized eta invariant η h h i ( e D ) take in general? Are theyalways algebraic numbers? IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 45
In particular, if a delocalized eta invariant is transcendental, then it will lead to a coun-terexample to the Baum-Connes conjecture [BC, BCH, C]. Note that the above question isa reminiscent of Atiyah’s question concerning rationality of ℓ -Betti numbers [A1]. Atiyah’squestion was answered in negative by Austin, who showed that ℓ -Betti numbers can betranscendental [Au].So far, we have been assuming the conjugacy class h h i has polynomial growth, whichguarantees the convergence of the integral in (10.1). In general, the integral in (10.1) fails toconverge. The following theorem of Chen-Wang-Xie-Yu [CWXY] gives a sufficient conditionfor when the integral in (10.1) converges. Theorem 10.4.
Let M be a closed manifold and f M the universal covering over M . Suppose D is a self-adjoint first-order elliptic differential operator over M and e D the lift of D to f M .If h h i is a nontrivial conjugacy class of π ( M ) and e D has a sufficiently large spectral gap atzero, then the delocalized eta invariant η h h i ( e D ) of e D is well-defined. We would like to emphasis that the theorem above works for all fundamental groups. Inthe special case where the conjugacy class h h i has sub-exponential growth, then any nonzerospectral gap is in fact sufficiently large, hence in this case η h h i ( e D ) is well-defined as long as e D is invertible.Let us make precise of what “sufficiently large spectral gap at zero” means. Fix a finitegenerating set S of Γ. Let ℓ be the corresponding word length function on Γ determined by S . Since S is finite, there exist C and K h h i > { g ∈ h h i : ℓ ( g ) = n } Ce K h h i · n . (10.2)We define τ h h i to be τ h h i = lim inf g ∈h h i ℓ ( g ) →∞ (cid:16) inf x ∈ f M dist( x, gx ) ℓ ( g ) (cid:17) . (10.3)Since the action of Γ on f M is free and cocompact, we have τ h h i > D by σ D ( x, v ), for x ∈ M and cotangent vector v ∈ T ∗ x M . We define the propagation speed of D to be the positive number c D = sup {k σ D ( x, v ) k : x ∈ M, v ∈ T ∗ x M, k v k = 1 } . Definition 10.5.
With the above notation, let us define σ h h i := 2 K h h i · c D τ h h i . (10.4)Recall that e D is said to have a spectral gap at zero if there exists an open interval ( − ε, ε ) ⊂ R such that spectrum( e D ) ∩ ( − ε, ε ) is either { } or empty. Moreover, e D is said to have a sufficiently large spectral gap at zero if its spectral gap is larger than σ h h i .Again it is natural to ask the following question. Open Question . With e D as in the above theorem, what values can the delocalized etainvariant η h h i ( e D ) take in general? Are they always algebraic numbers?A special feature of traces is that they always have uniformly bounded representatives,when viewed as degree zero cyclic cocycles. In fact, our proof of Theorem 10 . . C Γ has adecomposition respect to the conjugacy classes of Γ ([Nis]): HC ∗ ( C Γ) ∼ = Y h h i HC ∗ ( C Γ , h h i ) , where HC ∗ ( C Γ , h h i ) denotes the component that corresponds to the conjugacy class h h i . If h h i is a nontrivial conjugacy class, then a cyclic cocycle in HC ∗ ( C Γ , h h i ) will be called adelocalized cyclic cocycle at h h i . Theorem 10.7.
Assume the same notation as in Theorem . . Let ϕ be a delocalizedcyclic cocycle at a nontrivial conjugacy class h h i . If ϕ has exponential growth and e D hasa sufficiently large spectral gap at zero, then η ϕ ( e D ) is well-defined, where η ϕ ( e D ) is a higheranalogue ( cf. [CWXY]) of the formula (10.1) . For higher degree cyclic cocycles, the precise meaning of “sufficiently large spectral gap atzero” is similar to but slightly different from that of the case of traces. We refer the readerto [CWXY, Section 3.2] for more details. For now, we simply point out that if both Γ and ϕ have sub-exponential growth, then any nonzero spectral gap is in fact sufficiently large, hencein this case η ϕ ( e D ) is well-defined as long as e D is invertible. The explicit formula for η ϕ ( e D ) isdescribed in terms of the transgression formula for the Connes-Chern character [C, C2]. It isessentially a periodic version of the delocalized part of Lott’s noncommutative-differentialhigher eta invariant. We shall call η ϕ ( e D ) a delocalized higher eta invariant from now on.Formally speaking, just as Lott’s delocalized eta invariant η h h i ( e D ) can be interpreted asthe pairing between the degree zero cyclic cocycle tr h h i and the higher rho invariant ρ ( e D ),so can the delocalized higher eta invariant η ϕ ( e D ) be interpreted as the pairing between thecyclic cocycle ϕ and the higher rho invariant ρ ( e D ). A key analytic difficulty here is to verifywhen such a pairing is well-defined, or more ambitiously, to verify when one can extendthis pairing to a pairing between the cyclic cohomology of C Γ and the K -theory group K ∗ ( C ∗ L, ( f M ) Γ ). The group K ∗ ( C ∗ L, ( f M ) Γ ) consists of C ∗ -algebraic secondary invariants; inparticular, it contains all higher rho invariants from the discussion above. Such an extensionof the pairing is important, often necessary, for many interesting applications to geometryand topology (cf. [PS1, XY1, WXY]). We refer the reader to [CWXY] for details on how to identify the formula for η ϕ ( e D ) in Theorem 10 . IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 47
In [CWXY], such an extension of the pairing, that is, a pairing between delocalized cycliccocycles of all degrees and the K -theory group K ∗ ( C ∗ L, ( f M ) Γ ) was established, in the case ofGromov’s hyperbolic groups. More precisely, we have the following theorem [CWXY]. Theorem 10.8.
Let M be a closed manifold whose fundamental group Γ is hyperbolic. Sup-pose h h i is non-trivial conjugacy class of Γ . Then every element [ α ] ∈ HC k +1 − i ( C Γ , h h i ) induces a natural map τ [ α ] : K i ( C ∗ L, ( f M ) Γ ) → C such that the following are satisfied: ( i ) τ [ Sα ] = τ [ α ] , where S is Connes’ periodicity map S : HC ∗ ( C Γ , h h i ) → HC ∗ +2 ( C Γ , h h i );( ii ) if D is an elliptic operator on M such that the lift e D of D to the universal cover f M of M is invertible, then we have τ [ α ] ( ρ ( e D )) = η [ α ] ( e D ) , where ρ ( e D ) is the higher rho invariant of e D and η [ α ] ( e D ) is the delocalized higher etainvariant from Theorem . . In particular, in the case of hyperbolic groups, the delo-calized higher eta invariant η [ α ] ( e D ) is always well-defined, as long as e D is invertible. The construction of the map τ [ α ] in the above theorem uses Puschnigg’s smooth densesubalgebra for hyperbolic groups [P1] in an essential way. In more conceptual terms, theabove theorem provides an explicit formula to compute the delocalized Connes-Chern char-acter of C ∗ -algebraic secondary invariants. More precisely, the same techniques developedin [CWXY] actually imply that there is a well-defined delocalized Connes-Chern character Ch deloc : K i ( C ∗ L, ( f M ) Γ ) → HC deloc ∗ ( B ), where B is Puschnigg’s smooth dense subalgebra of C ∗ r (Γ) and HC deloc ∗ ( B ) is the delocalized part of the cyclic homology of B . Now for Gro-mov’s hyperbolic groups, every cyclic cohomology class of C Γ continuously extends to cycliccohomology class of B (cf. [P1] for the case of degree zero cyclic cocycles and [CWXY] for thecase of higher degree cyclic cocycles). Thus the map τ [ α ] can be viewed as a pairing betweencyclic cohomology and delocalized Connes-Chern characters of C ∗ -algebraic secondary in-variants. As a consequence, this unifies Higson-Roe’s higher rho invariant and Lott’s highereta invariant for invertible operators. In fact, even more is true. One can use the same techniques developed in [CWXY] to show thatif A is smooth dense subalgebra of C ∗ r (Γ) for any group Γ (not necessarily hyperbolic) and in addition A is a Fr´echet locally m -convex algebra, then there is a well-defined delocalized Connes-Chern character Ch deloc : K i ( C ∗ L, ( f M ) Γ ) → HC deloc ∗ ( A ). Of course, in order to pair such a delocalized Connes-Chern char-acter with a cyclic cocycle of C Γ, the key remaining challenge is to continuously extend this cyclic cocycleof C Γ to a cyclic cocycle of A . Here the definition of cyclic homology of B takes the topology of B into account, cf. [C2, Section II.5]. We point out that the proof of Theorem 10 . not rely on the Baum-Connes iso-morphism for hyperbolic groups [L, MY], although the theorem is closely connected to theBaum-Connes conjecture and the Novikov conjecture. On the other hand, if one is willingto use the full power of the Baum-Connes isomorphism for hyperbolic groups, there is infact a different, but more indirect, approach to the delocalized Connes-Chern character map.First, observe that the map τ [ α ] factors through a map τ [ α ] : K i ( C ∗ L, ( E Γ) Γ ) ⊗ C → C where E Γ is the universal space for proper Γ-actions. Now the Baum-Connes isomor-phism µ : K G ∗ ( E Γ) ∼ = −−→ K ∗ ( C ∗ r (Γ)) for hyperbolic groups implies that one can identify K i ( C ∗ L, ( E Γ) Γ ) ⊗ C with L h h i6 =1 HC ∗ ( C Γ , h h i ), where HC ∗ ( C Γ , h h i ) is the delocalized cyclichomology at h h i and the direct sum is taken over all nontrivial conjugacy classes. In partic-ular, after this identification, it follows that the map τ [ α ] becomes the usual pairing betweencyclic cohomology and cyclic homology. However, for a specific element, e.g. the higher rhoinvariant ρ ( e D ), in K i ( C ∗ L, ( E Γ) Γ ), its identification with an element in L h h i6 =1 HC ∗ ( C Γ , h h i )is rather abstract and implicit. More precisely, the computation of the number τ [ α ] ( ρ ( e D ))essentially amounts to the following process. Observe that if a closed spin manifold M isequipped with a positive scalar curvature metric, then stably it bounds (more precisely, theuniversal cover f M of M becomes the boundary of another Γ-manifold, after finitely manysteps of cobordisms and vector bundle modifications). In principle, the number τ [ α ] ( ρ ( e D ))can be derived from a higher Atiyah-Patodi-Singer index theorem for this bounding manifold.Again, there is a serious drawback of such an indirect approach — the explicit formula for τ [ α ] ( ρ ( e D )) is completely lost. In contrast, a key feature of the construction of the delocalizedConnes-Chern character map in Theorem 10 . C ∗ -algebraic secondary invariants. Their approach is in spirit similar to the indirectmethod just described above (making use of the Baum-Connes isomorphism for hyperbolicgroups), although their actual technical implementation is different.As an application, we use this delocalized Connes-Chern character map from Theorem10 . W be a compact n -dimensional spin manifold with boundary ∂W . Suppose W is equipped with a Riemannian metric g W which has product structure near ∂W and in addition has positive scalar curvature on ∂W . Let f W be the universal coveringof W and g f W the Riemannian metric on f W lifted from g W . With respect to the metric g f W , the associated Dirac operator e D W on f W naturally defines a higher index Ind Γ ( e D W ) (asin Section 7) in K n ( C ∗ ( f W ) Γ ) = K n ( C ∗ r (Γ)), where Γ = π ( W ). Since the metric g f W haspositive scalar curvature on ∂ f W , it follows from the Lichnerowicz formula that the associatedDirac operator e D ∂ on ∂ f W is invertible, hence naturally defines a higher rho invariant ρ ( e D ∂ ) IGHER INVARIANTS IN NONCOMMUTATIVE GEOMETRY 49 in K n − ( C ∗ L, ( f W ) Γ ). We have the following delocalized higher Atiyah-Patodi-Singer indextheorem. Theorem 10.9.
With the notation as above, if
Γ = π ( W ) is hyperbolic and h h i is a non-trivial conjugacy class of Γ , then for any [ ϕ ] ∈ HC ∗ ( C Γ , h h i ) , we have Ch [ ϕ ] ( Ind Γ ( e D W )) = − η [ ϕ ] ( e D ∂ ) , (10.5) where Ch [ ϕ ] ( Ind Γ ( e D W )) is the Connes-Chern pairing between the cyclic cohomology class [ ϕ ] and the higher index class Ind Γ ( e D W ) .Proof. This follows from Theorem 10 . . (cid:3) By using Theorem 10 .
8, we have derived Theorem 10 . K -theoreticcounterpart. This is possible only because we have realized η [ ϕ ] ( e D ∂ ) as the pairing betweenthe cyclic cocycle ϕ and the C ∗ -algebraic secondary invariant ρ ( e D ∂ ) in K ( C ∗ L, ( f W ) Γ ).Alternatively, one can also derive Theorem 10 . C ∗ r (Γ); ornoncommutative differential forms on a certain class of smooth dense subalgebras (if exist) ofgeneral C ∗ -algebras (not just group C ∗ -algebras) in Wahl’s version. In the case of Gromov’shyperbolic groups, one can choose such a smooth dense subalgebra to be Puschnigg’s smoothdense subalgebra B . As mentioned before, for Gromov’s hyperbolic groups, every cycliccohomology class of C Γ continuously extends to a cyclic cohomology class of B (cf. [P1]for the case of degree zero cyclic cocycles and [CWXY] for the case of higher degree cycliccocycles). Now Theorem 10 . C Γ.One can also try to pair the higher Atiyah-Patodi-Singer index formula of Leichtnam-Piazza and Wahl with group cocycles of Γ, or equivalently cyclic cocycles in HC ∗ ( C Γ , h i ),where h i stands for the conjugacy class of the identity element of Γ. In this case, forfundamental groups with property RD, Gorokhovsky, Moriyoshi and Piazza proved a higherAtiyah-Patodi-Singer index theorem for group cocycles with polynomial growth [Gr]. References [AD] G.
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Department of Mathematics, Texas A&M University, College Station, TX77843, USA
E-mail address : [email protected] (Guoliang Yu) Department of Mathematics, Texas A&M University, College Station, TX77843, USA, Shanghai Center for Mathematical Sciences, Shanghai, China
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