Homology, Homotopy and Applications

Volume 19 (2017)

Number 1

Remarks about $\Delta$-complexes and applications

Pages: 89 – 110

DOI: http://dx.doi.org/10.4310/HHA.2017.v19.n1.a5

Authors

Michael Pors (Department of Mathematics and Statistics, University of Calgary, Alberta, Canada)

Soumen Sarkar (Department of Mathematics and Statistics, University of Calgary, Alberta, Canada)

Peter Zvengrowski (Department of Mathematics and Statistics, University of Calgary, Alberta, Canada)

Abstract

We first consider some generalities regarding $\Delta$-complexes, in particular, we give a brief history of the category of $\Delta$-complexes, and its relation to the category of semi-simplicial complexes introduced in 1950 by Eilenberg and Zilber. A natural construction of $\Delta$-complexes arising from a group action on a simplicial complex is next considered. Finally, an application of this construction to obtain an elementary explicit computation of the cohomology ring $H^{*} (\mathbb{R}P^n ; \mathbb{Z}_2)$, based on a $\Delta$-complex structure, is given.

Keywords

$\Delta$-complex, EZ-complex, group action

2010 Mathematics Subject Classification

Primary 55N35. Secondary 57Q91.

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