The Free Loop Space Homology of $(n-1)$-connected $2n$-manifolds

TL;DR

This paper computes the integral free loop space homology of certain highly connected even-dimensional manifolds, extending known results and providing insights into the Batalin-Vilkovisky operator's action.

Contribution

It introduces methods to compute free loop space homology for a broad class of $(n-1)$-connected $2n$-manifolds, including cases with trivial cup product squares.

Findings

Computed homology for manifolds with specific connectivity and dimension.

Provided partial results on Batalin-Vilkovisky operator action.

Extended techniques to a wider class of manifolds.

Abstract

Our goal in this paper is to compute the integral free loop space homology of $(n-1)$-connected $2n$-manifolds $M$, $n\geq 2$. We do this when $n\neq 2,4,8$, or when $n\neq 2$ and $\tilde H^*(M)$ has trivial cup product squares, though the techniques used here should extend to a much wider range of manifolds. We also give partial information concerning the action of the Batalin-Vilkovisky operator.