I would like to learn something about $\mathbb{A}^1$-homotopy theory. I know about standard references on the subject, but before dwelling into studying them I have a doubt which some expert could clear.

Having a look at Levine's survey it seems that algebraic cobordism has been very succesful at solving open problems in $\mathbb{A}^1$-homotopy theory.

How much the two topics are independent of each other? Is it possible to understand the full implications of $\mathbb{A}^1$-homotopy theory without knowing about algebraic cobordism? Has the latter theory in some sense superseded the former?


1 Answer 1


The two topics are logically, if not morally, independent of one another. $\mathbb{A}^1$-homotopy encodes objects like motivic cohomology & it's relatives which are of interest regardless of the framework. There's no way for algebraic cobordism to supersede that -- algebraic cobordism is more directly comparable to Chow theory and $K^0$ than to motivic cohomology.

Conversely, algebraic cobordism provides a more geometric viewpoint on (a piece of) $MGL$ -- surely a valuable thing to have around as well as being of independent geometric interest. That said, if your interests are more motivic than geometric, you could get by without knowing the details of algebraic cobordism provided that you know all the classical statements in complex cobordism that inspired it.

Motivic vs algebraic cobordism.

The $\mathbb{P}^1$-spectrum $MGL$, or "motivic cobordism," enjoys a privileged role in the world of $\mathbb{A}^1$-homotopy similar to that of $MU$ in classical homotopy. There is a relationship between this "motivic cobordism" and "algebraic cobordism." The former is a bigraded theory, and Levine showed that $MGL^{2n,n}(X) = \omega^n(X)$. (This bigrading issue is analogous to how Chow theory occurs as the $(2n,n)$-graded piece of motivic cohomology, and explains why you don't get a long exact sequence in algebraic cobordism, etc...)

So one can view Morel-Levine's (or Levine-Pandharipande's) algebraic cobordism as giving an axiomatic (or geometric) viewpoint on the motivic theory, like we had for $MU$. Unlike the case of complex cobordism, where one can directly compare it to $MU$-cohomology using transversality results, here the comparison is much more difficult and computational. The proof of this comparison relies on a (currently unpublished) spectral sequence due to Hopkins-Morel. It should be noted that constructing this spectral sequence is hard, and by the time you've constructed it you've had to independently check lots of things that you might've wanted to deduce from the comparison with algebraic cobordism (for instance you pretty much end up computing $MGL^{2*,*}(Spec k)$, you can see the comparison of cobordism to Chow theory, etc.).

Degree formula (or, application to B-K)

The reference to open problems likely refers to the use of cobordism and Rost's degree formula in the final steps of proving Bloch-Kato for $\ell \neq 2$. Cobordism is a tool in the proof, but introducing algebraic cobordism is not strictly necessary. (One can get by with explicit computations with the characteristic numbers of interest. It'd certainly be fair to like the Levine-Morel proof of the degree formula, though.)

The Bloch-Kato conjecture is concerned with the "Galois symbol" map $$ K^n_M(k)/\ell = H^n(k, \mathbb{Z}/\ell(n)) \to H_{et}^n(k, \mathbb{Z}/\ell(n)) = H_{et}^n(k, \mu_\ell^{\otimes n}) $$ No cobordism in sight yet. Suslin-Merkurjev's proof for $n=2$ and Voevodsky's proof for $\ell=2$ made use of "splitting varieties" that one could write down pretty much explicitly and then proceed to study: Brauer-Severi varieties and Pfister quadrics, respectively. This doesn't seem to work for the general case, and instead one writes down a minimalist wishlist for splitting varieties and then has to show that they exist --- it is in this step where cobordism (or really, characteristic numbers) play a role.

A "splitting variety" for a non-zero symbol $0 \neq u = u_1 \otimes \cdots \otimes u_n \in K^n_M(k)/\ell$ should be a smooth variety $X/k$ such that $u$ pulls back to zero in $H^n(k(X), \mathbb{Z}/\ell(n))$ ("$X$ splits $u$"), with $\dim X = \ell^{n-1}-1$, and some more technical conditions, including a partial "universality" for this property: $X'$ splits $u$ iff there is a rational map from (a degree prime-to-$\ell$ cover of) $X'$ to $X$; it follows that $X$ must have no degree prime-to-$\ell$ zero-cycles (or else $Spec k$ splits $u$, i.e., $u = 0$).

Cobordism (of whatever flavor you like: complex cobordism suffices) enters when relating this to characteristic numbers: namely, to the property of being a $v_n$-variety (=representing a $v_n$ class in complex or algebraic cobordism, up to decomposables). Here, one needs something like "Rost's degree formula", which implies for instance that the property of being a $v_n$-variety with no prime-to-$\ell$ zero-cycles is invariant under prime-to-$\ell$ degree covers.


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