Algebraic & Geometric Topology

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

Man Gao and Jie Wu

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N91: Equivariant homology and cohomology [See also 19L47] 55P35: Loop spaces
Secondary: 55T05: General 55U10: Simplicial sets and complexes

homology decomposition loop space simplicial group group actions


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.

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