KSp-characteristic classes determine Spin$^h$ cobordism
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915814300450816 |
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| author | Buchanan, Jonathan McKean, Stephen |
| author_facet | Buchanan, Jonathan McKean, Stephen |
| contents | A classic result of Anderson, Brown, and Peterson states that the cobordism spectrum MSpin (respectively, MSpin$^c$) splits as a sum of Eilenberg--Mac Lane spectra and connective covers of real K-theory (respectively, complex K-theory) at 2. We develop a theory of symplectic K-theory classes and use these to build an explicit splitting for MSpin$^h$ in terms of Eilenberg--Mac Lane spectra and spectra related to symplectic K-theory. This allows us to determine the Spin$^h$ cobordism groups systematically. We also prove that two Spin$^h$-manifolds are cobordant if and only if their underlying unoriented manifolds are cobordant and their KSp-characteristic numbers agree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08209 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | KSp-characteristic classes determine Spin$^h$ cobordism Buchanan, Jonathan McKean, Stephen Algebraic Topology Primary: 19L41. Secondary: 53C27 A classic result of Anderson, Brown, and Peterson states that the cobordism spectrum MSpin (respectively, MSpin$^c$) splits as a sum of Eilenberg--Mac Lane spectra and connective covers of real K-theory (respectively, complex K-theory) at 2. We develop a theory of symplectic K-theory classes and use these to build an explicit splitting for MSpin$^h$ in terms of Eilenberg--Mac Lane spectra and spectra related to symplectic K-theory. This allows us to determine the Spin$^h$ cobordism groups systematically. We also prove that two Spin$^h$-manifolds are cobordant if and only if their underlying unoriented manifolds are cobordant and their KSp-characteristic numbers agree. |
| title | KSp-characteristic classes determine Spin$^h$ cobordism |
| topic | Algebraic Topology Primary: 19L41. Secondary: 53C27 |
| url | https://arxiv.org/abs/2312.08209 |