7 D ec 1 99 8 ALGEBRAIC INVARIANTS FOR HOMOTOPY TYPES

0 70 7 . 02 50 v 4 [ m at h . A T ] 1 8 D ec 2 00 8 THE REFINED TRANSFER , BUNDLE STRUCTURES AND ALGEBRAIC K - THEORY

We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen’s functor A(X). We also give concrete examples of f...

ar X iv : m at h / 97 12 25 1 v 2 [ m at h . G T ] 1 4 A ug 1 99 8 HOMOTOPY TYPES OF COMPLEMENTS OF 2 - ARRANGEMENTS IN R 4

We study the homotopy types of complements of arrangements of n transverse planes in R, obtaining a complete classification for n ≤ 6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the c...

ar X iv : d g - ga / 9 71 20 20 v 1 3 0 D ec 1 99 7 Weinstein Conjecture and GW Invariants

Lectures on the Cohomology of Groups

The cohomology theory of groups arose from both topological and algebraic sources. The starting point for the topological aspect of the theory was a 1936 paper by Hurewicz [7], in which he introduced aspherical spaces. These are spaces X such that πn(X) = 0 for n 6= 1. (Hurewicz had introduced higher homotopy groups just one year earlier, and he was now trying to understand the spaces with the ...

ar X iv : m at h / 99 04 03 1 v 1 [ m at h . G T ] 8 A pr 1 99 9 INVARIANTS OF GENERIC IMMERSIONS AND BERNOULLI NUMBERS

First order invariants of generic immersions of manifolds of dimension nm−1 into manifolds of dimension n(m+1)−1, m, n > 1 are constructed using the geometry of self-intersections. The range of one of these invariants is related to Bernoulli numbers. As by-products some geometrically defined invariants of regular homotopy are found.