Strong Novikov Conjecture Research Articles

Asymptotically flat Fredholm bundles and assembly

Almost flat finitely generated projective Hilbert [Formula: see text]-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a [Formula: see text]-homology class [Formula: see text] twisted with an almost flat bundle in terms of the image of [Formula: see text] under Lafforgue’s assembly map and the almost representation associated with the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to generalize Dadarlat’s theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.

The Novikov conjecture and extensions of coarsely embeddable groups

Let 1\to N\to G\to G/N\to 1 be a short exact sequence of countable discrete groups and let B be any G - C^* -algebra. In this paper, we show that the strong Novikov conjecture with coefficients in B holds for such a group G when the normal subgroup N and the quotient group G/N are coarsely embeddable into Hilbert spaces. As a result, the group G satisfies the Novikov conjecture under the same hypothesis on N and G/N .

$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

The main result of this paper is the $G$-homotopy invariance of the $G$-index of signature operator of proper co-compact $G$-manifolds. If proper co-compact $G$ manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the images of their signature operators by the $G$-index map are the same in the $K$-theory of the $C^{*}$-algebra of the group $G$. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required, so this is a generalization of the classical case of closed manifolds. Using this result we can deduce the equivariant version of Novikov conjecture for proper co-compact $G$-manifolds from the Strong Novikov conjecture for $G$.

Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra holds for many other exotic group C*-algebras, in particular the one associated to the smallest strongly Morita compatible and exact crossed product functor used in the new version of the Baum-Connes conjecture. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

The Baum–Connes conjecture localised at the unit element of a discrete group

AbstractWe construct a Baum–Connes assembly map localised at the unit element of a discrete group$\Gamma$. This morphism, called$\mu _\tau$, is defined in$KK$-theory with coefficients in$\mathbb {R}$by means of the action of the idempotent$[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$canonically associated to the group trace of$\Gamma$. We show that the corresponding$\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of$\mu _\tau$is functorial with respect to the group$\Gamma$.

Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras

Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

Banach strong Novikov conjecture for polynomially contractible groups

We prove the Banach strong Novikov conjecture for groups having polynomially bounded higher-order combinatorial functions. This includes all automatic groups.

On Banach spaces with Kasparov and Yu's Property (H)

A new geometric property of Banach spaces recently introduced by G. Kasparov and G. Yu, called Property (H), has important applications to the strong Novikov conjecture and the coarse Novikov conjecture, yet is far from being well understood. In this paper, we investigate various uniformly continuous maps on unit spheres of Banach spaces and prove that all separable Banach lattices with nontrivial cotype have Property (H).

The Novikov conjecture on Cheeger spaces

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L^2 -de Rham cohomology theory satisfying Poincaré duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mischenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl.

A finite-dimensional approach to the strong Novikov conjecture

The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using the Atiyah-Singer families index theorem, and partly by Carlsson's deformation $K$--theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the $K$--theory and cohomology of representation spaces.

Some ‘homological’ properties of the stable Higson corona

We establish certain ‘homological properties’ of the stable Higson corona used by Emerson and Meyer to study the Dirac-dual-Dirac approach to the Baum–Connes conjecture [5]. These are used to obtain explicit isomorphisms between the K-theory groups of stable Higson coronas, and the K-theory groups of certain geometrically defined boundaries. This is sufficient to give a simple proof of the strong Novikov conjecture for torsion-free hyperbolic groups and torsion-free groups acting properly and cocompactly on CAT(0) spaces, and also provides an input into an index theorem in single operator theory [15], [16].

The Novikov conjecture and geometry of Banach spaces

In this paper, we prove the strong Novikov conjecture for groups coarsely embeddable into Banach spaces satisfying a geometric condition called Property (H).

On Gromov’s scalar curvature conjecture

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group.0.10.1.Theorem.Suppose that a discrete groupπ\pihas the following properties:11. The Strong Novikov Conjecture holds forπ\pi.22. The natural mapper:kon(Bπ)→KOn(Bπ)per:ko_n(B\pi )\to KO_n(B\pi )is injective. Then the Gromov Macroscopic Dimension Conjecture holds true for spinnn-manifoldsMMwith the fundamental groupsπ1(M)\pi _1(M)that containπ\pias a finite index subgroup.

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF ∞ , i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF ∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes–Moscovici and Druţu–Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF ∞ and isocohomological.

The strong Novikov conjecture for low degree cohomology

We show that for each discrete group G, the rational assembly map K_*(BG) \otimes Q \to K_*(C*_{max} G) \otimes \Q is injective on classes dual to the subring generated by cohomology classes of degree at most 2 (identifying rational K-homology and homology via the Chern character). Our result implies homotopy invariance of higher signatures associated to these cohomology classes. This consequence was first established by Connes-Gromov-Moscovici and Mathai. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our previous work. In contrast to the argument of Mathai, our approach is independent of (and indeed gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature.

Geometrization of the Strong Novikov Conjecture for residually finite groups

In this paper, we prove that the Strong Novikov Conjecture for a residually finite group is essentially equivalent to the Coarse Geometric Novikov Conjecture for a certain metric space associated to the group. As an application, we obtain the Coarse Geometric Novikov Conjecture for a large class of sequences of expanders.

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups. We observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We combine this observation with known results to prove the Strong Novikov Conjecture for some exotic groups constructed by Osin.

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov on signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain higher Todd genera are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the strong Novikov Conjecture for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space

Corollary 1.2. Let Γ be a finitely generated group. If Γ, as a metric space with a word-length metric, admits a uniform embedding into Hilbert space, and its classifying space BΓ has the homotopy type of a finite CW complex, then the strong Novikov conjecture holds for Γ, i.e. the index map from K∗(BΓ) to K∗(C∗ r (Γ)) is injective. Corollary 1.2 follows from Theorem 1.1 and the descent principle [23]. By index theory, the strong Novikov conjecture implies the Novikov conjecture on the homotopy invariance of higher signatures (cf. [8] for an excellent