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Nielsen coincidence theory for $(n,1)$-valued pairs

Karel Dekimpe, Lore De Weerdt

Abstract

We generalise Nielsen theory to coincidences of pairs $(f,g)$ where $f:X\multimap Y$ is $n$-valued multimap and $g:X\to Y$ is a single-valued map, for $X$ and $Y$ closed oriented triangulable manifolds of equal dimension. We prove a Wecken theorem in this setting, and formulas for the Nielsen, Lefschetz and Reidemeister numbers in terms of the analogous invariants for single-valued maps. If $X$ and $Y$ are orientable infra-nilmanifolds, we derive explicit formulas in terms of the fundamental group morphisms of $f$ and $g$.

Nielsen coincidence theory for $(n,1)$-valued pairs

Abstract

We generalise Nielsen theory to coincidences of pairs where is -valued multimap and is a single-valued map, for and closed oriented triangulable manifolds of equal dimension. We prove a Wecken theorem in this setting, and formulas for the Nielsen, Lefschetz and Reidemeister numbers in terms of the analogous invariants for single-valued maps. If and are orientable infra-nilmanifolds, we derive explicit formulas in terms of the fundamental group morphisms of and .

Paper Structure

This paper contains 29 sections, 43 theorems, 231 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Coincidence theory
  3. Root theory
  4. Fixed point theory of $n$-valued maps
  5. Coincidence and root theory of $n$-valued maps
  6. The coincidence problem for $(n,1)$-valued pairs
  7. Overview of the paper
  8. Definitions
  9. Coincidence classes
  10. The splitting space of $f$
  11. The coincidence index
  12. The Nielsen number
  13. Reidemeister and Lefschetz number
  14. Two classical results
  15. Reduction to finitely many coincidences
  16. ...and 14 more sections

Key Result

Theorem 1.1

If $X$ is a compact manifold of dimension at least $3$, then $N(f)=\mathop{\mathrm{MF}}\nolimits[f]$ for any map $f:X\to X$.

Figures (4)

  • Figure 1: The coincidence class $C_t$ of $(f_t,g_t)$ can be visualised as the 'slice at $t$' of the coincidence class $C$ of $(F,G)$. If $V$ is an isolating neighborhood of $C$ in $\mathop{\mathrm{Coin}}\nolimits(F,G)$, then the slice at $t$ of $V$ is an isolating neighborhood of $C_t$ in $\mathop{\mathrm{Coin}}\nolimits(f_t,g_t)$.
  • Figure 2: A finite cover for $C_t\times \{t\}$ consisting of basic open subsets of $V$, and the corresponding sets $U_t$, $W'_t$ and $W_t$.
  • Figure 3: Idea of the proof of Theorem \ref{['thm:Wecken']}.
  • Figure :

Theorems & Definitions (103)

  • Theorem 1.1: Wecken Theorem
  • Example 1.2
  • Lemma 1.3
  • proof
  • Theorem 1.4
  • proof
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 93 more