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Non-affine $n$-valued maps on tori

Karel Dekimpe, Lore De Weerdt

Abstract

In this paper we construct $n$-valued maps on $k$-dimensional tori, where $n,k\geq 2$, that are not homotopic to affine $n$-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.

Non-affine $n$-valued maps on tori

Abstract

In this paper we construct -valued maps on -dimensional tori, where , that are not homotopic to affine -valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.
Paper Structure (5 sections, 16 theorems, 65 equations)

This paper contains 5 sections, 16 theorems, 65 equations.

Table of Contents

  1. Preliminaries on $n$-valued maps
  2. The divisibility condition
  3. The cycle condition
  4. The influence of the choice of lift
  5. Constructing more complicated non-affine maps

Key Result

Theorem 1.1

Given a map $f:X\to D_n(X)$, if $f_1:X\to D_m(X)$ and $f_2:X\to D_{n-m}(X)$ are maps such that $f=\{f_1,f_2\}$, the Nielsen number of $f$ is given by

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 23 more