Wednesday, 20 June 2018

Karel Dekimpe: Fixed points, Twisted Conjugacy and Lie Algebras

Let $G$ be a group and $\varphi:G \to G$ be an endomorphism. We will say that two elements $x$ and $y$ are twisted conjugate if $x= z y \varphi(z)^{-1}$ for some $z\in G$. Note that when $\varphi$ is the identity on $G$, the notion of twisted conjugacy reduces to that of ordinary conjugacy. The number of twisted conjugacy classes is called the Reidemeister number of $\varphi$ and is denoted by $R(\varphi)$.

Now, let $f:X \to X$ be a self map of a closed manifold $X$. In this talk I will explain how the number of fixed points of $f$ is related to the number of twisted conjugacy classes $R(f_\ast)$. Here $f_\ast$ is the endomorphism of the fundamental group of $X$ induced by $f$. Finally, I will show how Lie algebra techniques can be used to easily compute those Reidemeister numbers.

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