#### Volume 13, issue 6 (2013)

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

### Man Gao and Jie Wu

Algebraic & Geometric Topology 13 (2013) 3175–3201
##### 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.

##### Keywords
homology decomposition, loop space, simplicial group, group actions
##### Mathematical Subject Classification 2010
Primary: 55N91, 55P35
Secondary: 55T05, 55U10
##### Publication
Received: 8 January 2013
Revised: 7 April 2013
Accepted: 8 April 2013
Published: 11 September 2013
##### Authors
 Man Gao Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117542 Republic of Singapore Jie Wu Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117542 Republic of Singapore http://www.math.nus.edu.sg/~matwujie