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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 1

### Remarks about $\Delta$-complexes and applications

Pages: 89 – 110

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

#### Authors

#### 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.

Published 6 June 2017