## Advanced Studies in Pure Mathematics

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

### A universal bivariant theory and cobordism groups

#### Abstract

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

**Dates**

Received: 7 March 2008

Revised: 30 October 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543448026

**Digital Object Identifier**

doi:10.2969/aspm/05610363

**Mathematical Reviews number (MathSciNet)**

MR2604091

**Zentralblatt MATH identifier**

1189.14005

**Subjects**

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]

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aspm/1543448026