An elementary proof of the homotopy invariance of stabilized configuration spaces

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 ${\mathrm{\Sigma }}_{+}^{\mathrm{\infty }}F(M,k)$ 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.