## Algebraic & Geometric Topology

### 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 $C2$ 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 $C2$–invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the $C2$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3175-3201.

Dates
Revised: 7 April 2013
Accepted: 8 April 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715729

Digital Object Identifier
doi:10.2140/agt.2013.13.3175

Mathematical Reviews number (MathSciNet)
MR3248730

Zentralblatt MATH identifier
06213056

#### Citation

Gao, Man; Wu, Jie. Homology decompositions of the loops on 1–stunted Borel constructions of $C_2$–actions. Algebr. Geom. Topol. 13 (2013), no. 6, 3175--3201. doi:10.2140/agt.2013.13.3175. https://projecteuclid.org/euclid.agt/1513715729

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