A bi-variant algebraic cobordism via correspondences
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909212929425408 |
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| author | Yokura, Shoji |
| author_facet | Yokura, Shoji |
| contents | A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Ω^{*,\sharp}(X, Y)$ such that $Ω^{*,\sharp}(X, pt)$ is isomorphic to Lee--Pandharipande's algebraic cobordism of vector bundles $Ω_{-*,\sharp}(X)$. In particular, $Ω^{*}(X, pt)=Ω^{*, 0}(X, pt)$ is isomorphic to Levine--Morel's algebraic cobordism $Ω_{-*}(X)$. Namely, $Ω^{*,\sharp}(X, Y)$ is \emph{a bi-variant vesion} of Lee--Pandharipande's algebraic cobordism of bundles $Ω_{*,\sharp}(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_00269 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A bi-variant algebraic cobordism via correspondences Yokura, Shoji Algebraic Geometry Algebraic Topology Category Theory A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Ω^{*,\sharp}(X, Y)$ such that $Ω^{*,\sharp}(X, pt)$ is isomorphic to Lee--Pandharipande's algebraic cobordism of vector bundles $Ω_{-*,\sharp}(X)$. In particular, $Ω^{*}(X, pt)=Ω^{*, 0}(X, pt)$ is isomorphic to Levine--Morel's algebraic cobordism $Ω_{-*}(X)$. Namely, $Ω^{*,\sharp}(X, Y)$ is \emph{a bi-variant vesion} of Lee--Pandharipande's algebraic cobordism of bundles $Ω_{*,\sharp}(X)$. |
| title | A bi-variant algebraic cobordism via correspondences |
| topic | Algebraic Geometry Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2207.00269 |