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Posts Tagged ‘fixed point theorems’

The goal of this post is to compute the cohomology of the n-torus X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is \mathbb{Z} by default. At the end we will interpret this computation in terms of cohomology operations.

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The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let G be a finite p-group acting on a finite set X. Let X^G denote the subset of X consisting of those elements fixed by G. Then |X^G| \equiv |X| \bmod p; in particular, if p \nmid |X| then G has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

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