Advanced Studies in Pure Mathematics

A universal bivariant theory and cobordism groups

Shoji Yokura

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This is a survey on a universal bivariant theory $\mathbb{M}_{\mathcal{S}}^{\mathcal{C}} (X \to Y)$, which is a prototype of a bivariant analogue of Levine–Morel's algebraic cobordism, and its application to constructing a bivariant theory $F\Omega (X \to Y)$ of cobordism groups. Before giving such a survey, we recall the genus such as signature, which is the main important invariant defined on the cobordism group, i.e, a ring homomorphism from the cobordism group to a commutative ring with a unit. We capture the Euler–Poincaré characteristic and genera as a drastic generalization of the very natural counting of finite sets.

Article information

Singularities — Niigata–Toyama 2007, J.-P. Brasselet, S. Ishii, T. Suwa and M. Vaquie, eds. (Tokyo: Mathematical Society of Japan, 2009), 363-394

Received: 7 March 2008
Revised: 30 October 2008
First available in Project Euclid: 28 November 2018

Permanent link to this document euclid.aspm/1543448026

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Zentralblatt MATH identifier

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14C40: Riemann-Roch theorems [See also 19E20, 19L10] 14F99: None of the above, but in this section 19E99: None of the above, but in this section 55N35: Other homology theories 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90]

Bivariant theory bordism cobordism Borel–Moore functor Euler–Poincaré characteristic genus


Yokura, Shoji. A universal bivariant theory and cobordism groups. Singularities — Niigata–Toyama 2007, 363--394, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05610363.

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