Year of Publication

2020

Degree Name

Doctor of Philosophy (PhD)

Document Type

Doctoral Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Kate Ponto

Abstract

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 for describing the minimum set of fixed points for continuous maps of tori. That is, if $T$ is the $d$ dimensional torus and $f\colon T \to T$ is continuous, then there exists a map $g$ homotopic to $f$ so that \[|\Fix(g)|=|L(f)|\] A point $x\in T$ is a periodic point of order $n$ if $f^n(x)=x$. In this paper we realize the minimum set of periodic points of an endomorphism of tori by studying the sequence of Lefschetz numbers for the iterates of $f$, $\{L(f^m)\}_{m|n}$. More specifically, there exists a map $g$ homotopic to $f$ so that \[|\Fix(g)|=\sum_{m\vert n}(-1)^m\left| L(f^m) \right|.\] Furthermore, provided the sequence $\{L(f^m)\}_{m|n}$ is not identically zero, it also provides a complete lower bound for maps of tori parameterized by $S^1$. Therefore, if $F\colon S^1 \to \Endo(T)$ satisfies $L(F_x)\neq0$ for all $x\in S^1$, then $F\simeq G$ so that \[|\Fix(G_x)|=\sum_{m\vert n}(-1)^m\left| L(F_x^m) \right|. \] \newpage This behavior is very particular to maps of tori and is not expected to generalize to endomorphisms of other spaces, even manifolds. In fact, our extra requirement that $L(F_x)\neq 0$ for $F\colon S^1\to \Endo(T)$ is evidence for the requirements of the parameterized invariant suggested in \cite{MalkiewichPonto2018PeriodicPA}.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2020.233

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