Die Dachabbildung in ganzzahliger Cech-Homologie

TL;DR

This paper generalizes the symmetric squaring construction in Cech homology from Z/2-coefficients to integer coefficients for even dimensions, providing detailed proofs and exploring foundational aspects of Cech homology.

Contribution

It introduces the 'Dachabbildung' construction in integer Cech homology, extending previous Z/2-coefficients results and establishing its key properties with comprehensive proofs.

Findings

Generalization of symmetric squaring to integer coefficients for even dimensions

Proofs confirming the properties of the Dachabbildung construction

Analysis showing the equivalence of two definitions of Cech homology

Abstract

Looking at the cartesian product of a topological space with itself, a natural map to be considered on that object is the involution that interchanges the coordinates, i.e. that maps (x,y) to (y,x). The so-called halfsquaring construction, now also called "symmetric squaring construction", in Cech homology with Z/2-coefficients was introduced in [arXiv:0709.1774] as a map from the k-th Cech homology group of a space X to the 2k-th Cech homology group of X \times X divided by the above mentioned involution. It turns out to be a crucial construction in the proof of a Borsuk-Ulam-type theorem. In this thesis, a generalization of this construction, which is here called "Dachabbildung", to Cech homology with integer coefficients is given for even dimensions k, which is proven to equally satisfy the useful properties of the original construction. Detailed proofs are provided which could also supersede the somewhat sketchy exposition of [arXiv:0709.1774] for the case of Z/2-coefficients. As a preparation for the presented results, two possibilities of defining Cech homology are closely examined and shown to be ismorphic.