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Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle

Bartira Maués

TL;DR

This work develops a braid-theoretic framework to compute the Nielsen fixed point number for $2$-valued non-split maps on the Klein bottle by linking such maps to lift-factors and the Borsuk-Ulam Property through covering spaces. It extends Nielsen coincidence theory to non-orientable manifolds and provides explicit, braid-based formulas for $N(oldsymbol{ extphi})$ in terms of lift data, distinguishing two lift-type families (A and B) and detailing their associated covers and lifts. The results yield concrete expressions for the Klein bottle, expressed via $B_2(K)$-braid data and torus-to-Klein-bottle coincidences, and include a broader discussion of potential generalizations to $n$-valued maps and Wecken-type questions. Overall, the paper advances practical computation of Nielsen numbers for multivalued maps on non-orientable surfaces and clarifies the role of lift-factors and BU-property in this setting.

Abstract

In this paper we study 2-valued non-split maps, focusing on the Klein bottle. We establish a connection between a 2-valued non-split map $φ:X\multimap Y$ and a pair of classes of maps $([f],[f\circ δ])\in [\tilde X,Y]\times[\tilde X, Y]$, where $δ$ is a free involution on $\tilde X$, $X=\tilde X/δ$ and the class of the lift factor $[f]$ does not satisfy the Borsuk-Ulam Property in respect to $δ$. We also exhibit a method to compute the Nielsen fixed point number of a 2-valued non-split map on a closed connected manifold in terms of the Nielsen coincidence number between a lift factor and a covering space map, generalizing the formula from only orientable manifolds to also non-orientable manifolds. Finally we display a formula for the Nielsen fixed point number of 2-valued non-split maps on the Klein bottle in terms of two braids of the Klein bottle.

Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle

TL;DR

This work develops a braid-theoretic framework to compute the Nielsen fixed point number for -valued non-split maps on the Klein bottle by linking such maps to lift-factors and the Borsuk-Ulam Property through covering spaces. It extends Nielsen coincidence theory to non-orientable manifolds and provides explicit, braid-based formulas for in terms of lift data, distinguishing two lift-type families (A and B) and detailing their associated covers and lifts. The results yield concrete expressions for the Klein bottle, expressed via -braid data and torus-to-Klein-bottle coincidences, and include a broader discussion of potential generalizations to -valued maps and Wecken-type questions. Overall, the paper advances practical computation of Nielsen numbers for multivalued maps on non-orientable surfaces and clarifies the role of lift-factors and BU-property in this setting.

Abstract

In this paper we study 2-valued non-split maps, focusing on the Klein bottle. We establish a connection between a 2-valued non-split map and a pair of classes of maps , where is a free involution on , and the class of the lift factor does not satisfy the Borsuk-Ulam Property in respect to . We also exhibit a method to compute the Nielsen fixed point number of a 2-valued non-split map on a closed connected manifold in terms of the Nielsen coincidence number between a lift factor and a covering space map, generalizing the formula from only orientable manifolds to also non-orientable manifolds. Finally we display a formula for the Nielsen fixed point number of 2-valued non-split maps on the Klein bottle in terms of two braids of the Klein bottle.
Paper Structure (9 sections, 19 theorems, 52 equations, 3 figures)

This paper contains 9 sections, 19 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminairies on 2-valued maps and a connection to the Borsuk-Ulam Property
  3. Nielsen number for 2-valued non-split maps on connected closed manifolds
  4. The braid group on 2 strands on the Klein bottle
  5. Total Braid Group
  6. Nielsen number for 2-valued non-split maps on the Klein bottle
  7. Maps of type (A)
  8. Maps of type (B)
  9. Final comments

Key Result

Theorem 2.2

Let $\phi:X\multimap Y$ be a $2$-valued non-split map with correspondent map $\Phi:X\rightarrow D_2(Y)$, let $q:\tilde{X} \rightarrow X$ be the covering space induced by $\Phi_\#^{-1}(P_2(Y))$ and $\hat{\Phi}:\tilde{X}\rightarrow F_2(Y)$ a lift of $q\circ\Phi$. Let furthermore $\delta$ be the unique

Figures (3)

  • Figure 1: $\tilde{\mathbb T}$ and deck-transformation $\delta_A:\tilde{\mathbb T}\rightarrow \tilde{\mathbb T}$
  • Figure 2: path between $p_1$ and $p_2$ in $\mathbb K$
  • Figure 3: $\tilde{\mathbb K}$ and deck-transformation $\delta_B$

Theorems & Definitions (34)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • Remark 4.3
  • Proposition 4.4
  • Proposition 4.5
  • ...and 24 more