Some remarks on configuration spaces

Raptis, George

doi:10.1090/S0002-9939-2010-10580-4

Some remarks on configuration spaces
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by George Raptis

Proc. Amer. Math. Soc. 139 (2011), 1879-1887

DOI: https://doi.org/10.1090/S0002-9939-2010-10580-4

Published electronically: October 6, 2010

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Abstract:

This paper studies the homotopy type of the configuration spaces $F_n(X)$ by introducing the idea of configuration spaces of maps. For every map $f: X \to Y$, the configuration space $F_n(f)$ is the space of configurations in $X$ that have distinct images in $Y$. We show that the natural maps $F_n(X) \leftarrow F_n(f) \rightarrow F_n(Y)$ are homotopy equivalences when $f$ is a proper cell-like map between $d$-manifolds. We also show that the best approximation to $X \mapsto F_n(X)$ by a homotopy invariant functor is given by the $n$-fold product map.

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