  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.
