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.