A twisted homology fibration criterion and the twisted group-completion theorem

TL;DR

This paper clarifies and generalizes the homology fibration criterion and group-completion theorem to twisted coefficients, aiding the analysis of the homology of oriented configuration spaces in manifolds.

Contribution

It provides a detailed correction and extension of existing theorems to include twisted coefficients, with applications to configuration space homology.

Findings

Generalized homology fibration criterion to twisted coefficients

Refined the group-completion theorem for twisted homology

Applied results to oriented configuration spaces

Abstract

The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem and to generalize both this and the homology fibration criterion of McDuff to homology with twisted coefficients. This will be used in forthcoming work to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.