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Homological data on the periodic structure of self-maps on wedge sums

Marcos J. González, Víctor F. Sirvent, Richard Urzúa

TL;DR

The paper develops a homological framework for self-maps on wedge sums of compact manifolds, expressing Lefschetz numbers, zeta functions, and Dold coefficients in terms of coordinate maps via a squared-by-blocks decomposition. It proves general reduction theorems that reduce invariants of a wedge-sum map to those of its coordinate maps, and applies these results to wedge sums of tori to reveal obstructions and nontrivial periodic-point structure. Through explicit examples, it demonstrates realizability conditions for coordinate data and analyzes algebraic periods, offering conjectures and open questions that connect homology with dynamical periodicity. The work provides practical formulas enabling computation of fixed-point and periodicity information for a broad class of maps, with potential implications for dynamics on ENRs and polyhedra.

Abstract

In this article, we study the periodic points for continuous self-maps on the wedge sum of topological manifolds, exhibiting a particular combinatorial structure. We compute explicitly the Lefschetz numbers, the Dold coefficients and consider its set of algebraic periods. Moreover, we study the special case of maps on the wedge sum of tori, and show some of the homological obstructions present in defining these maps.

Homological data on the periodic structure of self-maps on wedge sums

TL;DR

The paper develops a homological framework for self-maps on wedge sums of compact manifolds, expressing Lefschetz numbers, zeta functions, and Dold coefficients in terms of coordinate maps via a squared-by-blocks decomposition. It proves general reduction theorems that reduce invariants of a wedge-sum map to those of its coordinate maps, and applies these results to wedge sums of tori to reveal obstructions and nontrivial periodic-point structure. Through explicit examples, it demonstrates realizability conditions for coordinate data and analyzes algebraic periods, offering conjectures and open questions that connect homology with dynamical periodicity. The work provides practical formulas enabling computation of fixed-point and periodicity information for a broad class of maps, with potential implications for dynamics on ENRs and polyhedra.

Abstract

In this article, we study the periodic points for continuous self-maps on the wedge sum of topological manifolds, exhibiting a particular combinatorial structure. We compute explicitly the Lefschetz numbers, the Dold coefficients and consider its set of algebraic periods. Moreover, we study the special case of maps on the wedge sum of tori, and show some of the homological obstructions present in defining these maps.

Paper Structure

This paper contains 6 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statements of the main results
  3. Proofs of the main results and applications
  4. Self-maps on wedge sums of tori
  5. Examples
  6. Open questions and further remarks

Key Result

Theorem 1

Let $f\colon \bigvee_{i=1}^sX_i \to \bigvee_{i=1}^sX_i$ be a continuous permutative squared by blocks self-map with associated permutation $\sigma$. Let $s=s_1+s_2+\cdots+s_q$ be the partition associated with the cyclic representation of $\sigma$. Then, for any partition $\Lambda_1,\Lambda_2, \ldots

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 11 more