Symmetric Squaring in Homology and Bordism

Abstract

Looking at the cartesian product X \times X of a topological space X 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 'symmetric squaring construction' in Čech homology with Z/2-coefficients was introduced by Schick et al. 2007 as a map from the k-th Čech homology group of a space X to the 2k-th Čech 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 parametrised Borsuk-Ulam Theorem. The symmetric squaring construction can be generalized to give a map in bordism, which will be the main topic of this thesis. More precisely, it will be shown that there is a well-defined, natural map from the k-th singular bordism group of X to the 2k-th bordism group of X \times X divided by the involution as above. Moreover, this squaring really is a generalisation of the Čech homology case since it is compatible with the passage from bordism to homology via the fundamental class homomorphism. On the way to this result, the concept of Čech bordism is first defined as a combination of bordism and Čech homology and then compared to Čech homology.