Manifold Aspects of the Novikov Conjecture 1975

be the Hirzebruch L-class of an oriented manifold M. Let B (or K(; 1)) denote any aspherical space with fundamental group. (A space is aspherical if it has a contractible universal cover.) In 1970 Novikov made the following conjecture. Many surveys have been written on the Novikov Conjecture. The goal here is to give an old-fashioned point of view, and emphasize connections with characteristic classes and the topology of manifolds. For more on the topology of manifolds and the Novikov Conjecture see 58], 47], 17]. This article ignores completely connections with C-algebras (see the articles of Mishchenko, Kasparov, and Rosenberg in 15]), applications of the Novikov conjecture (see 58],,9]), and most sadly, the beautiful work and mathematical ideas uncovered in proving the Novikov Conjecture in special cases (see 14]). The level of exposition in this survey starts at the level of a reader of Milnor-Stashee's book Characteristic Classes, but by the end demands more topological prerequisites. Here is a table of contents: Partially supported by the NSF. This survey is based on lectures given in Mainz, Germany in the Fall of 1993. The author wishes to thank the seminar participantsas well as Paul Kirk, Chuck McGibbon, and Shmuel Weinberger for clarifying conversations. 1 Does this refer to smooth, PL, or topological manifolds? Well, here it doesn't really matter. If the Novikov Conjecture is true for all smooth manifolds mapping to B, then it is true for all PL and topological manifolds mapping to B. However, the deenition of L-classes for topological manifolds depends on topological transversality 25], which is orders of magnitude more diicult than transversality for smooth or PL-manifolds. The proper category of manifolds will be a problem of exposition throughout this survey.