A geometric computation of cohomotopy groups in co-degree one
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912646228344832 |
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| author | Jung, Michael Rot, Thomas O. |
| author_facet | Jung, Michael Rot, Thomas O. |
| contents | Using geometric arguments, we compute the group of homotopy classes of maps from a closed $(n+1)$-dimensional manifold to the $n$-sphere for $n \geq 3$. Our work extends results from Kirby, Melvin and Teichner for closed oriented 4-manifolds and from Konstantis for closed $(n+1)$-dimensional spin manifolds, considering possibly non-orientable and non-spinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for the case that $n \geq 4$, we discuss applications for rank $n$ spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a non-vanishing section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03805 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A geometric computation of cohomotopy groups in co-degree one Jung, Michael Rot, Thomas O. Geometric Topology Algebraic Topology 55Q55 (Primary), 57R15, 57R22, 55N25 Using geometric arguments, we compute the group of homotopy classes of maps from a closed $(n+1)$-dimensional manifold to the $n$-sphere for $n \geq 3$. Our work extends results from Kirby, Melvin and Teichner for closed oriented 4-manifolds and from Konstantis for closed $(n+1)$-dimensional spin manifolds, considering possibly non-orientable and non-spinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for the case that $n \geq 4$, we discuss applications for rank $n$ spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a non-vanishing section. |
| title | A geometric computation of cohomotopy groups in co-degree one |
| topic | Geometric Topology Algebraic Topology 55Q55 (Primary), 57R15, 57R22, 55N25 |
| url | https://arxiv.org/abs/2307.03805 |