Homology Decompositions of the Loops on 1-Stunted Borel Constructions of C_2-Actions

Abstract: Carlsson's 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 this loop space has a mod 2 homology decomposition; ii) 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 completely compute the mod 2 Betti numbers for an example.