Steven C. Ferry

Novikov Conjectures, Index Theorems and Rigidity

This is an expository paper explaining coarse analogues of the Novikov Conjecture and describing how information on the original Novikov Conjecture can be derived from these. For instance, we will explain how Novikov’s theorem on the topological invariance of rational Pontrjagin classes is a consequence of a coarse theorem (whose proof we sketch in an appendix) that in turn also implies the Novikov Conjecture for nonpositively curved manifolds. We also formalize the technique so that it can be applied in a wide variety of other contexts. Thus, besides a few purely geometric results, we also discuss equivariant, A-theoretic, stratified, and foliated versions of the higher signature problem. Closely related papers are [GL], [CGM], [CP], [KaS], [HR], [Hu]. See also the surveys [We1], [FRW], for wider perspectives.

Topology of Homology Manifolds

ANR homology n-manifolds are nite-dimensional absolute neighborhood retracts X with the property that for every x 2 X, Hi(X;X fxg) is 0 for i 6= n and Z for i = n. Topological manifolds are natural examples of such spaces. To obtain nonmanifold examples, we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components. The resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group. It is a consequence of the double suspension theorem of Cannon and Edwards that, as in the examples above, the singularities of polyhedral ANR homology manifolds are isolated. There are, however, many examples of ANR homology manifolds which have no manifold points whatever. See [12] for a good exposition of the relevant theory. The purpose of this paper is to begin a surgical classi cation of ANR homology manifolds, sometimes referred to in the sequel, simply as homology manifolds. One way to approach this circle of ideas is via the problem of characterizing topological manifolds among ANR homology manifolds. In Cannons work on the double suspension problem [6], it became clear that in dimensions greater than 4, the right transversality hypothesis is the following (weak) form of general position.

Novikov Conjectures, Index Theorems and Rigidity: A history and survey of the Novikov conjecture

1. Precursors of the Novikov Conjecture 8 Characteristic classes 8 Geometric rigidity 9 The Hirzebruch signature theorem 9 The converse of the signature theorem (Browder, Novikov) 10 Topological invariance of the rational Pontrjagin classes (Novikov) 11 Non-simply-connected surgery theory (Novikov, Wall) 11 Higher signatures 12 Discovery of special cases of the Novikov Conjecture (Rokhlin, Novikov) 13 2. The Original Statement of the Novikov Conjecture 13 O nerexennyh zadaqah 13 [An English Version:] Unsolved Problems 15 3. Work related to the Novikov Conjecture: The First 12 Years or So 17 Statements of the Novikov and Borel Conjectures 17 Mishchenko and the symmetric signature 18 Lusztig and the analytic approach 20 Splitting theorems for polynomial extensions 21 Cappell and codimension 1 splitting theorems 22 Mishchenko and Fredholm representations 23 Farrell-Hsiang and the geometric topology approach 24 Kasparov and operator-theoretic K-homology 25 Surgery spectra and assembly (Quinn) 25 4. Work related to the Novikov Conjecture: The Last 12 Years or So, I:

Algebraic K-theory with continuous control at infinity

Let (E, Σ) be a pair of spaces consisting of a compact Hausdorff space Ē and a closed subspace Σ. Let U be an additive category. This paper introduces the category B(E, Σ; U of geometric modules over E with coefficients in U and with continuous control at infinity. One of the main results is to show that the functor that sends a CW complex X to the algebraic K-theory of B(cX, X; U) is a homology theory. Here cX is the closed cone on X and X is its base. The categories B(E, Σ; U) are generalizations of the categories C(Z; U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here (Z, ϱ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C(O(X); U)→B(cX, X; U). A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B(E, Σ; U) depends only on the topology of (E, Σ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

Topology of homology manifolds

We construct examples of nonresolvable generalized

Novikov conjectures, index theorems and rigidity : Oberwolfach 1993

n

The rational classification of links of codimension > 2

-manifolds,

Pro-excisive functors

n\geq 6

Limits of polyhedra in Gromov–Hausdorff space

, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed

A Survey of Bounded Surgery Theory and Applications

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