On the deleted squares of lens spaces

TL;DR

This paper investigates whether the homotopy equivalence of deleted squares of 3D lens spaces implies the spaces are homeomorphic, using differential characters and Massey products to analyze their topological differences.

Contribution

It provides new insights into the relationship between the homotopy types of deleted squares and the homeomorphism classes of lens spaces, addressing a specific open question.

Findings

Homotopy equivalence of deleted squares does not necessarily imply homeomorphism of lens spaces.

Differential characters and Massey products distinguish non-homeomorphic lens spaces with homotopy equivalent deleted squares.

The study advances understanding of the topological invariants related to lens spaces and their configuration 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.