The universal property of bordism of commuting involutions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913373666410496 |
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| author | Hausmann, Markus Schwede, Stefan |
| author_facet | Hausmann, Markus Schwede, Stefan |
| contents | We propose a formalism to capture the structure of the equivariant bordism rings of smooth manifolds with commuting involutions. We introduce the concept of an oriented el$_2^{RO}$-algebra, an algebraic structure featuring representation graded rings for all elementary abelian 2-groups, connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom class; this data is subject to one exactness property. Besides equivariant bordism, oriented global ring spectra also give rise to oriented el$_2^{RO}$-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el$_2^{RO}$-algebras are delocalized generalizations of global 2-torsion group laws.
Our main result shows that equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra. Several other interesting equivariant homology theories can also be characterized, on elementary abelian 2-groups, by similar universal properties. We prove that stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el$_2^{RO}$-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el$_2^{RO}$-algebra with an orientation that is both additive and invertible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The universal property of bordism of commuting involutions Hausmann, Markus Schwede, Stefan Algebraic Topology 55N22, 55N91, 55P91, 57R85 We propose a formalism to capture the structure of the equivariant bordism rings of smooth manifolds with commuting involutions. We introduce the concept of an oriented el$_2^{RO}$-algebra, an algebraic structure featuring representation graded rings for all elementary abelian 2-groups, connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom class; this data is subject to one exactness property. Besides equivariant bordism, oriented global ring spectra also give rise to oriented el$_2^{RO}$-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el$_2^{RO}$-algebras are delocalized generalizations of global 2-torsion group laws. Our main result shows that equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra. Several other interesting equivariant homology theories can also be characterized, on elementary abelian 2-groups, by similar universal properties. We prove that stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el$_2^{RO}$-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el$_2^{RO}$-algebra with an orientation that is both additive and invertible. |
| title | The universal property of bordism of commuting involutions |
| topic | Algebraic Topology 55N22, 55N91, 55P91, 57R85 |
| url | https://arxiv.org/abs/2406.00404 |