User markus land - MathOverflow

A formal group law over oriented bordism

2013-03-08T10:25:10Z

My question is related to the following question by Mark Grant here on math overflow:

http://mathoverflow.net/questions/74770/formal-group-law-of-unoriented-cobordism

There it is stated that $MO_*$ has a formal group law $F_0$, universal for formal group laws in characteristic $2$ which satisfy $F(x,x) = 0$.

Of course this formal group law is classified by a map $MU_* \to MO_*$ and in view of the model of $MU$ as unitary ring spectrum and $MO$ as orthogonal ring spectrum this map is indeed induced by a canonical map of (unitary) ring spectra $$c: MU \to MO,$$

see also the answer by Neil Strickland here: http://mathoverflow.net/questions/55602/which-cohomology-theories-are-real-and-complex-orientable .

Now I have two questions:

Q1) Does the map $c$ factor over $MSO$ ?

Q2) If so, can one say anything about the corresponding formal group law over $MSO$?

In view of how I think one can construct the map $MU \to MO$ I think it should be obvious that the map factors through $MSO$, since the map $U(n) \to O(2n)$ factors over $SO(2n)$. But I have not been able to find a reference that gives the example of $MSO$ being complex oriented.

So I guess Q2) is more interesting, already for the simple reason that the ring $MSO_\ast$ is much more delicate than the ring $MO_\ast$. Moreover (as Johannes Ebert also pointed out in a comment in the link above) real oriented theories come equipped with a map from $MO$ and hence split in Eilenberg-MacLane spectra, whereas $MSO$ behaves quite differently.

In addition, I have one last question:

Supposing Q1) and Q2) are true, then as I have already mentioned that $MSO$ is complex oriented. In particular we have a notion of abstract chern classes for $MSO$-cohomology. Moreover we have a canonical map $MSO \to H\mathbb{Z}$. The composite $MU \to H\mathbb{Z}$ is just the usual orientation of integral cohomolgy, in particular the abstract chern classes of this complex oriented theory are just the usual chern classes, but since oriented real bundles have pontryagin classes I ask myself:

Q3) Is there any way to view pontryagin classes as special case of abstract chern classes?

(It seems not to be the case for the cohomology theory $MSO$ which I would have thought of first).

stackification commutes with finite limits?

2012-11-12T14:40:10Z

Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes. Let us denote the category of stacks over $\mathcal{C}$ by $Stacks(\mathcal{C})$. This is naturally a full subcategory of the category $Pre_{grpds}(\mathcal{C})$ of presheaves in groupoids over $\mathcal{C}$. This is equivalent to formulating stacks as being categories fibered in groupoids, fulfilling effective descent.

Now the inclusion $Stacks(\mathcal{C}) \to Pre_{grpds}(\mathcal{C})$ has a left adjoint called the stackification functor.

It is a classical fact that the sheafification functor on presheaves of sets commutes with finite limits. Is this also true for the mentioned stackification functor?

Precisely, I want to know if and if so, why, the stackification functor commutes with 2-categorical pullbacks, here is the precise setting.

Suppose we are given Hopf-algebroids $P_1,P_2,P$ and view them as presheaves in groupoids. Suppose we are given two morphisms $P \to P_i$ and we construct the 2-pullback of functors $$Q = pullback(P_1 \to P \leftarrow P_2).$$

In my situation I can show that Q is equivalent to an affine scheme, and I want to conclude that the pullback of the stackified diagram is also equivalent to an affine scheme. If I knew that stackification commuted with finite limits, this would be ok, since the (co)unit (I always mix them up) of the adjunction is an equivalence, i.e., if $X$ is a stack, and $st(X)$ is its stackification, then the natural map $X\to st(X)$ adjoint to the identity of $st(X)$ is an equivalence of stacks.

Any help and comments are appreciated, also if you could give references concerning these questions, that would be great.

Comment by Markus Land

2013-03-17T10:31:29Z

I also believe that it is "well-known" that stable operations of degree $1$ are just given by Ext-groups of the two coefficient groups between which you want to calculate operations. Of course this implies that $H\mathbb{Z}^1H\mathbb{Z} = 0$.

Comment by Markus Land

2013-03-07T17:20:59Z

So this canonical map $MU \to MO$ will surely factor over $MSO$, right? Can one describe what formal group law it classifies? Do you know any references where this map is studied?

Comment by Markus Land

2012-11-14T22:58:40Z

I am not sure I know what you mean. Following Charles Rezk I wanted to define a stack over a site as a presheaf in groupoids such that it is a "homotopy sheaf" i.e. the functor evaluated on an object is given by a homotopy limit over a simplicial diagram induced by a covering in the site. Do you know what the splitting is in this viewpoint?