Products in spin$^c$-cobordism
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915575072030720 |
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| author | Abdallah, Hassan Salch, Andrew |
| author_facet | Abdallah, Hassan Salch, Andrew |
| contents | We calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $\leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10045 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Products in spin$^c$-cobordism Abdallah, Hassan Salch, Andrew Algebraic Topology 55N22, 53C27 We calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $\leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor. |
| title | Products in spin$^c$-cobordism |
| topic | Algebraic Topology 55N22, 53C27 |
| url | https://arxiv.org/abs/2407.10045 |