For a closed PL manifold M, we consider the configuration space F(M,k) of
ordered k-tuples of distinct points in M. We show that a suitable iterated
suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we
require depends on three parameters: the number of points k, the dimension of M
and the connectivity of M. Our proof uses a mixture of Poincare embedding
theory and fiberwise algebraic topology.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-35.abs.htm
The concept of thickening was systematically studied by C. T. C. Wall in [Wa1]. The suspension theorem of that paper is an exact sequence relating the n-dimensional thickenings of a finite complex to its ðn þ 1Þ-dimensional ones. The object of this note is to fill in what we believe is a missing argument in the proof of that theorem.
Fix K a finite connected CW complex of dimension ≤ k. An n-thickening of K is a pairin which M is a compact n-dimensional manifold and f : K → M is a simple homotopy equivalence. This concept was first introduced by C.T.C. Wall approximately 40 years ago. Most of the known results about thickenings are in a range of dimensions depending on k, n and the connectivity of K.In this paper we remove the connectivity hypothesis on K. We define moduli space of n-thickenings T n (K). We also define a suspension map E : T n (K) → T n+1 (K) and compute its homotopy fibers in a range depending only on n and k. We will show that these homotopy fibers can be approximated by certain section spaces whose definition depends only on the choice of a certain stable vector bundle over K.
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