同调与配边中的对称平方
Symmetric Squaring in Homology and Bordism
摘要：
考察一个拓扑空间X与其自身的笛卡儿积X \times X，在该对象上要考虑的自然映射是互换坐标的对合，即将（x，y）映射到（y，x）。Schick等人在2007年引入了Z/2系数的Čech同调中的所谓“对称平方构造”，作为从空间X的第k个Čech同调群到X的第2k个Čech同调群\times X除以上述对合的映射。它是证明参数化Borsuk-Ulam定理的一个重要构造。对称平方构造可以推广到边数映射，这是本文的主要内容。更精确地说，我们将证明存在一个从X的第k个奇异边群到X的第2k个边群的定义明确的自然映射\times X除以上面的对合。此外，这种平方实际上是Čech同调情形的推广，因为它与通过基本类同态从协边到同调的过程是相容的。在得到这个结果的过程中，我们首先将切赫边界主义的概念定义为边界主义和切赫同调的结合，然后将其与切赫同调进行比较。
英文摘要：
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.
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