# nLab algebraic cobordism

Contents

### Context

#### Cobordism theory

Concepts of cobordism theory

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:

algebraic:

# Contents

## 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

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 $X \in \Sm_k$.

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:

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 $X(b)^*(-)$ out of a pair consisting of a homotopy commutative H-monoid $X$ and a class $b\in \pi_n(X)$:

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

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

Then Snaith introduces a variant of such constructions with a more general ring $A$ replacing the complex numbers; and uses the Quillen’s description of algebraic K-theory of a ring $A$ in terms of the classifying space $B GL(A)$; 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 $M GL$ 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 $MGL^{*,*}$ 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

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

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