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Nielsen numbers of $n$-valued maps on infra-solvmanifolds

Karel Dekimpe, Lore De Weerdt

Abstract

We derive a formula for the Nielsen number $N(f)$ for every $n$-valued self-map $f$ of an infra-solvmanifold. To do this, we express $N(f)$ in terms of Nielsen coincidence numbers of single-valued maps on solvmanifolds, and derive a formula for Nielsen coincidence numbers in that setting.

Nielsen numbers of $n$-valued maps on infra-solvmanifolds

Abstract

We derive a formula for the Nielsen number for every -valued self-map of an infra-solvmanifold. To do this, we express in terms of Nielsen coincidence numbers of single-valued maps on solvmanifolds, and derive a formula for Nielsen coincidence numbers in that setting.

Paper Structure

This paper contains 18 sections, 23 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Nielsen theory
  3. Nielsen coincidence theory
  4. Nielsen fixed point theory of $n$-valued maps
  5. The structure of solvmanifolds
  6. Nilmanifolds
  7. Solvmanifolds
  8. The Mostow fibration
  9. Maps of special solvmanifolds
  10. Nielsen coincidence numbers on special solvmanifolds
  11. Coincidences on tori
  12. Coincidences of fiber maps
  13. Coincidences on solvmanifolds
  14. Nielsen numbers of $n$-valued maps on infra-solvmanifolds
  15. Infra-solvmanifolds
  16. ...and 3 more sections

Key Result

Theorem 2.3

Let $f:M\to D_n(M)$ be an $n$-valued map, and let $\varphi_j:S_j\to \pi$ be the morphisms induced by some lift $(\tilde{f}_1,\ldots,\tilde{f}_n):\tilde{M}\to F_n(\tilde{M},\pi)$ of $f$. Suppose $\bar{M}=\Gamma\backslash \tilde{M}$ is an orientable finite regular covering space of $M$, such that $\pi for all $\hat{x}\in\hat{M}$, and such that $\tilde{f}_j:\tilde{M}\to \tilde{M}$ is a lift of $\bar{

Theorems & Definitions (59)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Definition 3.4: see iris
  • Remark 3.5
  • ...and 49 more