Volume 4, issue 2 (2004)

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On the homotopy invariance of configuration spaces

Mokhtar Aouina and John R Klein

Algebraic & Geometric Topology 4 (2004) 813–827
 arXiv: math.AT/0310483
Abstract

For a closed PL manifold $M$, we consider the configuration space $F\left(M,k\right)$ of ordered $k$–tuples of distinct points in $M$. We show that a suitable iterated suspension of $F\left(M,k\right)$ 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 Poincaré embedding theory and fiberwise algebraic topology.

Keywords
configuration space, fiberwise suspension, embedding up to homotopy, Poincaré embedding
Mathematical Subject Classification 2000
Primary: 55R80
Secondary: 57Q35, 55R70