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algebraic cobordism

(Co)homological theories for algebraic varieties have led Grothendieck to propose a theory of motives which has partly been realized (e.g. in its triangulated and dg-versions) so far. One systematic approach is to study the homotopy theory in this context, namely ${𝔸}^1$-homotopy theory. However in practice, one is led to calculate invariants and it is difficult to find new computable cases; the study of algebraic cycles in higher codimension is notoriously difficult, and the Chow groups are difficult to compute. Somewhat more accessible is algebraic K-theory. This is unlike the classical situation where in addition many other extraordinary (co)homology theories are at hand. In the abstract setup, Voevodsky and others did study more general extraordinary cohomology theories, but down-to-Earth examples are difficult to find. This was roughly the motivation for two people – Marc Levine and Fabien Morel – to construct in a single effort the notion of algebraic cobordism, which is an analogue of complex cobordism for algebraic varieties. Many of the tools from Quillen’s and Novikov’s work on complex cobordism, like Lazard’s ring and formal groups, play a major role.

Algebraic cobordism is an axiomatically defined oriented Borel–Moore homology theory, where ‘oriented’ stands for the existence of correponding Chern classes. A very nontrivial theorem shows the existence of such a theory; and the theory became soon of significant practical use. The fact that ${𝔸}^1$-homotopy theory is not used —that is, the constructions are more direct— is an advantage.

On the other hand a universal oriented motivic cohomology theory is not the algebraic cobordism of Morel and Levine but rather the cohomology associated to the motivic spectrum? $\mathrm{MGL}$ of Voevodsky, which is different and also may be called a theory of algebaric cobordism. An early different approach to some sort of algebraic cobordism is from late 1970-s work of Victor Snaith, discussed in that entry.

The standard references are pretty readable and complete for a general algebraic geometry audience.

• M. Levine, F. Morel, Algebraic cobordism, Springer 2007.

• Marc Levine, Algebraic cobordism, Proceedings of the ICM, Beijing 2002, vol. 2, 57–66, math.KT/0304206

• M. Levine, F. 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)).

• M. Levine, R. Pandharipande, Algebraic cobordism revisited, math.AG/0605196

• M. Levine, F. Morel, Oberwolfach Arbeitsgemeinschaft mit aktuellem Thema, April 2005 programme and references.pdf

More recently, there is also an algebraic analogue of h-cobordism:

• Aravind Asok, Fabien Morel, Smooth varieties up to ${𝔸}^1$-homotopy and algebraic h-cobordisms, arXiv:0810.0324