A Twisted Homology Fibration Criterion and the Twisted Group-Completion Theorem

j.p chaplin;

The purpose of this note is to clarify some details in McDuff and Segal’s proof of the group completion theorem in McDuff and Segal [Homology fibrations and the ‘group-completion’ theorem, Invent. Math. 31 (1975/76), 279–284] and generalize this and the homology fibration
criterion of McDuff [Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107] to homology with twisted coefficients. This will be used in Miller and Palmer [Scanning for oriented configuration spaces, preprint, 2014] to identify the limiting homology of ‘oriented’ configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.