The Dyer-Lashof algebra in bordism (extended abstract)
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Bisson, Terrence Joyal, André |
| author_facet | Bisson, Terrence Joyal, André |
| contents | We present a theory of Dyer-Lashof operations in unoriented bordism (the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where $N_*( )$ is unoriented bordism and $H_*( )$ is homology mod 2, does not respect these operations). For any finite covering space we define a ``polynomial functor'' from the category of topological spaces to itself. If the covering space is a closed manifold we obtain an operation defined on the bordism of any $E_\infty$-space. A certain sequence of operations called squaring operations are defined from two-fold coverings; they satisfy the Cartan formula and also a generalization of the Adem relations that is formulated by using Lubin's theory of isogenies of formal group laws. We call a ring equipped with such a sequence of squaring operations a D-ring, and observe that the bordism ring of any free $E_\infty$-space is free as a D-ring. In particular, the bordism ring of finite covering manifolds is the free D-ring on one generator. In a second compte-rendu we discuss the (Nishida) relations between the Landweber-Novikov and the Dyer-Lashof operations, and show how to represent the Dyer-Lashof operations in terms of their actions on the characteristic numbers of manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Dyer-Lashof algebra in bordism (extended abstract) Bisson, Terrence Joyal, André Algebraic Topology 55N22 (Primary) We present a theory of Dyer-Lashof operations in unoriented bordism (the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where $N_*( )$ is unoriented bordism and $H_*( )$ is homology mod 2, does not respect these operations). For any finite covering space we define a ``polynomial functor'' from the category of topological spaces to itself. If the covering space is a closed manifold we obtain an operation defined on the bordism of any $E_\infty$-space. A certain sequence of operations called squaring operations are defined from two-fold coverings; they satisfy the Cartan formula and also a generalization of the Adem relations that is formulated by using Lubin's theory of isogenies of formal group laws. We call a ring equipped with such a sequence of squaring operations a D-ring, and observe that the bordism ring of any free $E_\infty$-space is free as a D-ring. In particular, the bordism ring of finite covering manifolds is the free D-ring on one generator. In a second compte-rendu we discuss the (Nishida) relations between the Landweber-Novikov and the Dyer-Lashof operations, and show how to represent the Dyer-Lashof operations in terms of their actions on the characteristic numbers of manifolds. |
| title | The Dyer-Lashof algebra in bordism (extended abstract) |
| topic | Algebraic Topology 55N22 (Primary) |
| url | https://arxiv.org/abs/2505.03115 |