Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910921564094464 |
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| author | Maués, Bartira |
| author_facet | Maués, Bartira |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle Maués, Bartira Algebraic Topology Primary 55M20, Secondary 20F36 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. |
| title | Computation of the Nielsen fixed point number for 2-valued non-split maps on the Klein bottle |
| topic | Algebraic Topology Primary 55M20, Secondary 20F36 |
| url | https://arxiv.org/abs/2504.20171 |