Complex K-theory of 4-complexes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913652931559424 |
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| author | Rosenberg, Jonathan |
| author_facet | Rosenberg, Jonathan |
| contents | This short note summarizes a number of facts about the ring $K^0(X)$ for $X$ a $4$-dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes, that every even cohomology class arises as a Chern class of a vector bundle, and that $K^0(X)$ is completely determined as a ring by knowledge of the even-dimensional cohomology ring $H^{\text{even}}(X; \mathbb Z)$. (All of these fail in high dimensions.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex K-theory of 4-complexes Rosenberg, Jonathan K-Theory and Homology Algebraic Topology 19L64 (Primary) 19L10, 55R50, 55R40 (Secondary) This short note summarizes a number of facts about the ring $K^0(X)$ for $X$ a $4$-dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes, that every even cohomology class arises as a Chern class of a vector bundle, and that $K^0(X)$ is completely determined as a ring by knowledge of the even-dimensional cohomology ring $H^{\text{even}}(X; \mathbb Z)$. (All of these fail in high dimensions.) |
| title | Complex K-theory of 4-complexes |
| topic | K-Theory and Homology Algebraic Topology 19L64 (Primary) 19L10, 55R50, 55R40 (Secondary) |
| url | https://arxiv.org/abs/2409.16557 |