Homology decompositions of the loops on 1–stunted Borel constructions of $C_2$–actions

Abstract

The Carlsson construction is a simplicial group whose geometric realization is the loop space of the 1–stunted reduced Borel construction. Our main results are: (i) given a pointed simplicial set acted upon by the discrete cyclic group $C_2$ of order 2, if the orbit projection has a section, then the loop space on the geometric realization of the Carlsson construction has a mod 2 homology decomposition; (ii) in addition, if the reduced diagonal map of the $C_2$–invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the $C_2$–invariant set and the orbit space. Result (i) generalizes a previous homology decomposition of the second author for trivial actions. To illustrate these two results, we compute the mod 2 Betti numbers of an example.