An elementary proof of the homotopy invariance of stabilized configuration spaces

Connor Malin

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arXiv · 2208.05947

An elementary proof of the homotopy invariance of stabilized configuration spaces

Connor Malin

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Abstract

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $\Sigma^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuration spaces'' which have recently undergone study.

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Connor Malin. 2022-08-11. An elementary proof of the homotopy invariance of stabilized configuration spaces. https://arxiv.org/abs/2208.05947

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