Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

A remark on the double complex of a covering for singular cohomology

Pages: 59 – 68

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a4

Authors

Roberto Frigerio (Dipartimento di Matematica, Università di Pisa, Italy)

Andrea Maffei (Dipartimento di Matematica, Università di Pisa, Italy)

Abstract

Given an open covering of a paracompact topological space $X$, there are two natural ways to construct a map from the cohomology of the nerve of the covering to the cohomology of $X$. One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the double complex associated to an open covering, and has a more algebraic flavour. In this paper we prove that these two maps coincide.

Keywords

nerve of a covering, double complex associated to an open covering

2010 Mathematics Subject Classification

55N05, 55N10, 55T99

Received 7 March 2020

Received revised 2 July 2020

Accepted 11 October 2020

Published 7 April 2021