Homology decompositions of the loops on 1–stunted Borel constructions of C2–actions

Gao, Man¹ ; Wu, Jie¹

¹Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117542, Republic of Singapore

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3175-3201

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.3175

Classification : 55N91, 55P35, 55T05, 55U10
Keywords: homology decomposition, loop space, simplicial group, group actions

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