Homotopy Invariance Of Higher Signatures Research Articles

The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

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.

Homotopy invariance of higher signatures and $3$-manifold groups

Nous demontrons que pour des varietes fermees et orientees les signatures qui proviennent des groupes fondamentaux d'une large classe de varietes orientables de dimension 3 sont des invariants homotopiques. Cette classe, que nous decrivons soigneusement, contient en particulier les varietes geometriques par morceaux au sens de Thurston. Si la conjecture de geometrisation de Thurston s'avere vraie cette classe coincide alors avec celle des groupes fondamentaux de varietes de dimension 3 orientables. Plus precisement nous demontrons que tous les groupes dans cette classe satisfont la conjecture de Baum-Connes avec coefficients. Nous discutons egalement le cas non-orientable.

On the cut-and-paste property of higher signatures of a closed oriented manifold

We extend the notion of the symmetric signature σ( M ̄ ,r)∈L n(R) for a compact n-dimensional manifold M without boundary, a reference map r : M→BG and a homomorphism of rings with involutions β : Z G→R to the case with boundary ∂M, where ( M ̄ , ∂M)→(M, ∂M) is the G-covering associated to r. We need the assumption that C ∗( ∂M) ⊗ Z G R is R-chain homotopy equivalent to a R-chain complex D ∗ with trivial mth differential for n=2 m resp. n=2 m+1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed oriented manifold with reference map Z→ BG. Let F⊂ Z be a cutting codimension one submanifold F⊂ Z and let F ̄ →F be the associated G-covering. Denote by α m( F ̄ ) the mth Novikov–Shubin invariant and by b m (2)( F ̄ ) the mth L 2-Betti number. If for the discrete group G the Baum–Connes assembly map is rationally injective, then we use σ( M ̄ ,r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have α m( F ̄ )=∞ + in the case n=2 m and, in the case n=2 m+1, if we have α m( F ̄ )=∞ + and b m (2)( F ̄ )=0 . This additivity result had been proved (by a different method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satisfied and additivity fails. We explain at the end of the introduction why our paper is greatly motivated by and partially extends some of the work of Leichtnam et al. (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999), Lott (Math. Ann., 1999) and Weinberger (Contemporary Mathematics, 1999, p. 231).

On the homotopy invariance of higher signatures for manifolds with boundary

If M is a compact oriented manifold-with-boundary whose fundamental group is virtually nilpotent or Gromov-hyperbolic, we show that the higher signatures of M are oriented-homotopy invariants. We give applications to the question of when higher signatures of closed manifolds are cut-and-paste invariant.

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

Higher eta invariants and the Novikov conjecture on manifolds with boundary

Let ( N,g) be a closed Riemannian manifold of dimension 2 m-1 and Γ → N ˜ → N be a Galois covering of N. We assume that Γ is of polynomial growth and that Δ N ˜ is L 2-invertible in degree m. By employing spectral sections which are symmetric with respect to the *-Hodge operator, we define the higher eta invariant associated to the signature operator on N ˜ , thus extending previous work of Lott. If π 1 ( M ) → M ˜ → M is the universal cover of a compact orientable even-dimensional manifold with boundary ( ∂M = N) then, under the above invertibility assumption on Δ ∂ M ˜ , we define a canonical Atiyah-Patodi-Singer signature-index class, in K 0 ( C r * (Γ)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above. We apply these results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary.

Cyclic cohomology, the Novikov conjecture and hyperbolic groups

(Received in revised form 27 April 1989) NOVIKOV'S conjecture on the homotopy invariance of higher signatures (28) can be formulated as follows: given a finitely presented group I and a compact oriented smooth manifold M, together with a continuous map +:M+BI, the generalized signatures (L(M)* $*(<), (Ml), where 5 runs over all classes in H*(BI, Q) and L(M) denotes the total Hirzebruch L-class of M, are homotopy invariants of the pair (M, $). In other words, if h: N+M is a hotiiotopy equivalence of oriented smooth manifolds, then (L(N)* h*($*(t)), (iV))=(L(M)*$*({), (Ml). The validity of this conjecture has been established, by a variety of techniques, for many groups I, most notably for closed discrete subgroups of finitely connected Lie groups. The latter result is due to Kasparov (24) and its proofs is based on bivariant K-theory. In this paper we present a new and more direct method for attacking the Novikov conjecture, which yields a proof of the conjecture for Gromov's (word) hyperbolic groups ( 183. These groups form an extremely rich and interesting class of finitely presented groups, which differs significantly, both in size and in nature, from the groups for which Novikov's conjecture was previously known. First of all, as pointed out by Gromov (18), they are generic among all finitely presented groups in the following sense: the ratio between the number of hyperbolic groups and all groups with a fixed number of generators and a fixed number of relations, each of length at most 1, tends to 1 when I+co (18,0.2(A)). Secondly, when adding at random relations to a (non-elementary) hyperbolic group, one obtains again a hyperbolic group (18,5.5). Thirdly, the cohomology of any finite polyhedron can be embedded into the cohomology of a hyperbolic group (18, 0.2(c)). Also, many of the hyperbolic groups exhibit exotic properties, like Kazhdan's property T ( 18, 5.61 or being non-linear (in a non-trivial way). Our approach is based on expressing the higher signatures in terms of the pairing between cyclic cohomology and K-theory (cf. (8)). The hyperbolicity assumption plays a twofold role: first, it ensures, via a deep result of Gromov (18; 8.3T-J, that every class < E Hk*' (BT, C) can be represented by a bounded group cocycle, and secondly, it enables us to make use of a critical norm estimate (first proved by Haagerup (20) for free groups), recently extended by Jolissaint (23) and de la Harpe (21) to hyperbolic groups. The paper is organized as follows. Using the Alexander-Spanier realization of the cohomology of a smooth manifold, reviewed in $1, we define in $2 localized analytic indices

On homotopy invariance of higher signatures

On homotopy invariance of higher signatures