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algebraic cobordism

Idea

Algebraic cobordism is the bigraded generalized cohomology theory represented by the motivic Thom spectrum $MGL$. Hence it is the algebraic or motivic analogue of complex cobordism. The $(2n,n)$-graded part has a geometric description via cobordism classes, at least over fields of characteristic zero.

Definition

Let $S$ be a scheme and $MGL_S$ the motivic Thom spectrum over $S$. Algebraic cobordism is the generalized motivic cohomology theory? $MGL_S^{*,*}$ represented by $MGL_S$:

… formula here …

Properties

The (2n,n)-graded part

Let $S = Spec(k)$ where $k$ is a field of characteristic zero. A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by Marc Levine and Fabien Morel. More precisely, Levine-Morel constructed the universal oriented cohomology theory $\Omega^* : \Sm_k \to CRing^*$. 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.

Theorem

(Levine-Morel). There is a canonical isomorphism of graded rings

$$\mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k))$$

where $\mathbf{L}^*$ denotes the Lazard ring with an appropriate grading.

Theorem

(Levine-Morel). Let $i : Z \hookrightarrow X$ be a closed immersion of smooth $k$-schemes and $j : U \hookrightarrow X$ the complementary open immersion. There is a canonical exact sequence of graded abelian groups

$$\Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0,$$

where $d = \codim(Z, X)$.

Theorem

(Levine-Morel). Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings

$$\Omega^*(k) \longrightarrow MU^{2*}(pt)$$

is invertible.

Theorem

(Levine 2008). The canonical homomorphisms of graded rings

$$\Omega^*(X) \longrightarrow MGL^{2*,*}(X)$$

are invertible for all $X \in \Sm_k$.

Related concepts

• motivic Thom spectrum
• complex cobordism
• motivic homotopy theory
• motivic cohomology
• algebraic K-theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\,M B$ (B-bordism):

• MFr

• MO, MSO, MSpin, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSp

relative bordism theories:

• MOFr, MUFr, MSUFr

equivariant bordism theory:

• equivariant MFr

• equivariant MO

• equivariant MU

global equivariant bordism theory:

• global equivariant mO,

• global equivariant mU

algebraic:

• algebraic cobordism

References

• Vladimir Voevodsky, $\mathbf{A}^1$-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 $MGL^{*,*}$ is considered in

• Marc Levine, Oriented cohomology, Borel-Moore homology and algebraic cobordism, arXiv.

The comparison with $MGL^{2*,*}$ 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

• Parker Lowrey, Timo Schuerg. Derived algebraic bordism, 2012, arXiv:1211.7023.

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 $\mathbb{A}^1$-homotopy and algebraic h-cobordisms, arXiv:0810.0324