Physical Sciences and Mathematics Commons^™

Periodic Points On Tori: Vanishing And Realizability, Shane Clark

Theses and Dissertations--Mathematics

Let $X$ be a finite simplicial complex and $f\colon X \to X$ be a continuous map. A point $x\in X$ is a fixed point if $f(x)=x$. Classically fixed point theory develops invariants and obstructions to the removal of fixed points through continuous deformation. The Lefschetz Fixed number is an algebraic invariant that obstructs the removal of fixed points through continuous deformation. \[L(f)=\sum_{i=0}^\infty (-1)^i \tr\left(f_i:H_i(X;\bQ)\to H_i(X;\bQ)\right). \] The Lefschetz Fixed Point theorem states if $L(f)\neq 0$, then $f$ (and therefore all $g\simeq f$) has a fixed point. In general, the converse is not true. However, Lefschetz Number is a complete invariant …