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

TL;DR

This paper develops homology decompositions for loop spaces of 1-stunted Borel constructions under C_2-actions, generalizing previous results and providing explicit computations of mod 2 Betti numbers.

Contribution

It introduces new homology decomposition results for loop spaces with C_2-actions, extending prior work to non-trivial actions and including explicit Betti number calculations.

Findings

Loop space homology decompositions under C_2-actions are established.

Generalization of previous trivial action results to non-trivial actions.

Explicit computation of mod 2 Betti numbers for an example.

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.