Die Dachabbildung in ganzzahliger Cech-Homologie

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.