Algebraic cobordism is the bigraded generalized cohomology theory represented by the motivic Thom spectrum . Hence it is the algebraic or motivic analogue of complex cobordism. The -graded part has a geometric description via cobordism classes, at least over fields of characteristic zero.
Let be a scheme and the motivic Thom spectrum over . Algebraic cobordism is the generalized motivic cohomology theory? represented by :
… formula here …
The (2n,n)-graded part
Let where is a field of characteristic zero. A geometric description of the -graded part of algebraic cobordism was given by Marc Levine and Fabien Morel. More precisely, Levine-Morel constructed the universal oriented cohomology theory . Here oriented signifies the existence of direct image or Gysin homomorphisms for proper morphisms of schemes. This implies the existence ofChern classes for vector bundles.
(Levine-Morel). There is a canonical isomorphism of graded rings
where denotes the Lazard ring with an appropriate grading.
(Levine-Morel). Let be a closed immersion of smooth -schemes and the complementary open immersion. There is a canonical exact sequence of graded abelian groups
(Levine-Morel). Given an embedding , the canonical homomorphism of graded rings
(Levine 2008). The canonical homomorphisms of graded rings
are invertible for all .
Vladimir Voevodsky, -Homotopy Theory, Doc. Math., Extra Vol. ICM 1998(I), 417-442, web.
Ivan Panin, K. Pimenov, Oliver Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology, Homotopy and Applications, 2008, 10(2), 211-226, arXiv.
Ivan Panin, K. Pimenov, Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Inventiones mathematicae, 2009, 175(2), 435-451, DOI, arXiv.
Markus Spitzweck, Algebraic cobordism in mixed characteristic, arXiv.
Marc Hoyois, From algebraic cobordism to motivic cohomology, pdf, arXiv.
Marc Levine, Girja Shanker Tripathi?, Quotients of MGL, their slices and their geometric parts, arXiv:1501.02436.
A discussion on MathOverflow:
- Interdependence between A^1-homotopy theory and algebraic cobordism, MO/36659.
The oriented cohomology theory
Marc Levine, Fabien Morel, Cobordisme algébrique I, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723–728, 2001 (doi01833-X)); Cobordisme algébrique II, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815–820, 2001 (doi01832-8)).
Marc Levine, Fabien Morel, Algebraic cobordism, Springer 2007, pdf.
Marc Levine, A survey of algebraic cobordism, pdf
Marc Levine, Algebraic cobordism, Proceedings of the ICM, Beijing 2002, vol. 2, 57–66, math.KT/0304206
Marc Levine, Three lectures on algebraic cobordism, University of Western Ontario Mathematics Department, 2005, Lecture I, Lecture II, Lecture III.
M. Levine, F. Morel, Oberwolfach Arbeitsgemeinschaft mit aktuellem Thema, April 2005 report, notes
A simpler construction was given in
- M. Levine, R. Pandharipande, Algebraic cobordism revisited, math.AG/0605196
A Borel-Moore homology? version of is considered in
- Marc Levine, Oriented cohomology, Borel-Moore homology and algebraic cobordism, arXiv.
The comparison with is in
- Marc Levine, Comparison of cobordism theories, Journal of Algebra, 322(9), 3291-3317, 2009, arXiv.
The construction was extended to derived schemes in the paper
The close connection of algebraic cobordism with K-theory is discussed in
- José Luis González, Kalle Karu. Universality of K-theory. 2013. arXiv:1301.3815.
An algebraic analogue of h-cobordism:
- Aravind Asok, Fabien Morel, Smooth varieties up to -homotopy and algebraic h-cobordisms, arXiv:0810.0324