Analytic Novikov for topologists

Abstract

Abstract. We explain for topologists the "dictionary" for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:(1) Why do the analytic proofs of the Novikov conjecture require the introduction of C * -algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of L-theory? Aren't they computing the wrong thing? (3) How can one show that the index map µ or β studied by operator theorists matches up with the assembly map in surgery theory? (4) Where does "bounded surgery theory" appear in the analytic proofs?Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?The literature on the Novikov conjecture (see [FRR]) consists of several different kinds of papers. Most of these fall into two classes: those based on topological arguments, usually involving surgery theory, and those based on analytic arguments, usually involving index theory. The purpose of this note is to "explain" the second class of papers to those familiar with the first class. I do not intend here to give a detailed sketch of the Kasparov KK-approach to the Novikov conjecture (for which the key details appear in [HsR]. Rather, I intend to concentrate on explaining the "dictionary" for relating the two main classes of papers on the Novikov conjecture, the topological and the analytic, and on trying

Equivariant Euler characteristics andK–homology Euler classes for proper cocompactG–manifolds

“…(Rationally, when G = {1}, Sign(M ) is the Poincaré dual of the total Lclass, the Atiyah-Singer L-class, which differs from the Hirzebruch L-class only by certain well-understood powers of 2, but in addition, it also carries quite interesting integral information [11], [22], [27]. A partial analysis of the class Sign G (M ) for G finite may be found in [26] and [24].…”

C*-Algebras and Controlled Topology

“…and we then follow this by Mischenko's natural map from L-theory to topological K-theory for C * -algebras [25,37].…”

“…Nevertheless it can be shown that the injectivity of assembly on the C * -algebra level implies (modulo 2-torsion) the injectivity of assembly on the L-theory level. All this is explained in the paper of Rosenberg [37], to which we also refer for an extensive bibliography on the Novikov conjecture.…”

Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups

“…The assumption that the action of Γ at the compactification EΓ is small allows us to regain control, i.e., to get a continuous control from a bounded control. See [Pe] and [Ro1] for details of such explanations of the proof in [CP3]. Remark 3.3 A related result is given by Ferry-Weinberger [FW1], where the boundary ∂EΓ of a compactification EΓ is required to be a Z-set, i.e., there exists a homotopy h t : ∂EΓ → EΓ, t ∈ [0, 1], such that h 0 is the identity map (or the inclusion), and h t (∂EΓ) ⊂ EΓ for t > 0; and EΓ is small in the sense that every continuous bounded map f : EΓ → EΓ extends by the identity map to a continuous map f : EΓ → EΓ.…”