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algebraic cobordism

Contents

Contents

Idea

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

Definition

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

… formula here …

Properties

The (2n,n)-graded part

Let S=Spec(k)S = Spec(k) where kk is a field of characteristic zero. A geometric description of the (2n,n)(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 Ω *:Sm kCRing *\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

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

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

Theorem

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

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

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

Theorem

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

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

is invertible.

Theorem

(Levine 2008). The canonical homomorphisms of graded rings

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

are invertible for all XSm kX \in \Sm_k.


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

bordism theory\;M(B,f) (B-bordism):

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

References

There are two notions of “algebraic cobordism”, not closely related, one due to Snaith 77, and one due to Levine-Morel 01.

Snaith’s construction

The construction in Snaith 77, motivated from the Conner-Floyd isomorphism, uses a variant of his general construction (Snaith's theorem) of a periodic multiplicative cohomology theory X(b) *()X(b)^*(-) out of a pair consisting of a homotopy commutative H-monoid XX and a class bπ n(X)b\in \pi_n(X):

When X=BS 1X = B S^1 (the classifying space of the circle group) and bb is a generator of π 2(BS 1)\pi_2(BS^1)\cong\mathbb{Z} then X(b) *()X(b)^*(-) is isomorphic with 2-periodic complex K-theory.

When X=BUX = B U and bb a generator of π 2(BU)\pi_2(BU)\cong\mathbb{Z} one obtains MU *[u 2,u 2 1]MU^*[u_2,u_2^{-1}] where MU is the (topological) complex cobordism cohomology and u 2u_2 is the periodicity element.

Then Snaith introduces a variant of such constructions with a more general ring AA replacing the complex numbers; and uses the Quillen’s description of algebraic K-theory of a ring AA in terms of the classifying space BGL(A)B GL(A); this way he obtains an algebraic cobordism theory.

Later, Gepner-Snaith 08 returned to the question of algebraic cobordism this time using the motivic version of algebraic cobordism of Voevodsky, namely the motivic spectrum MGLM GL representing universal oriented motivic cohomology theory (which is different from Morel-Voevodsky algebraic cobordism), and to the motivic version of Conner-Floyd isomorphism for which they give a comparably short proof.

Morel-Levine’s construction

More chat about the relation to motivic homotopy theory:

  • Interdependence between A^1-homotopy theory and algebraic cobordism, MO/36659.

A simpler construction was given in

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

A Borel-Moore homology version of MGL *,*MGL^{*,*} is considered in

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

The comparison with MGL 2*,*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

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

Last revised on February 28, 2021 at 09:00:39. See the history of this page for a list of all contributions to it.