An elementary proof of the homotopy invariance of stabilized configuration spaces

TL;DR

This paper provides an elementary proof that the equivariant stable homotopy type of configuration spaces on manifolds remains invariant under proper homotopies, extending to generalized configuration spaces.

Contribution

It introduces a simple proof method based on Spanier-Whitehead duality, avoiding complex machinery and allowing extension to generalized configuration spaces.

Findings

Proves homotopy invariance of configuration spaces using elementary techniques.

Extends the invariance result to generalized configuration spaces.

Provides a new approach that simplifies previous proofs.

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.