On the deleted squares of lens spaces

Abstract

The configuration space $F_2 (M)$ of ordered pairs of distinct points in a manifold $M$, also known as the deleted square of $M$, is not a homotopy invariant of $M$: Longoni and Salvatore produced examples of homotopy equivalent lens spaces $M$ and $N$ of dimension three for which $F_2 (M)$ and $F_2 (N)$ are not homotopy equivalent. In this paper, we study the natural question whether two arbitrary $3$-dimensional lens spaces $M$ and $N$ must be homeomorphic in order for $F_2 (M)$ and $F_2 (N)$ to be homotopy equivalent. Among our tools are the Cheeger--Simons differential characters of deleted squares and the Massey products of their universal covers.