Elliptic cohomology

This is part of an algebraic topology problem list, maintained by Mark Hovey.

Elliptic cohomology is quite important, so see the major problem list as well.

  1. Prove that the integral elliptic cohomology introduced by Hopkins admits an orientation from MO8, the 7-connected bordism spectrum. Ando, Hopkins, and Strickland thought they could do this at one point, and maybe they still can--I don't know.

  2. Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do with 2-categories. If you think about vector bundles, you have a bunch of local trivializations, tied together by these transition functions. The transition functions must satisfy a cocycle condition--here we are thinking of a vector bundle as a locally free sheaf, or some such nonsense. I mean on U the vector bundle is U cross R^n, on V it is V cross R^n, so on U intersect V you have two different trivializations, related by a transition function. Then on U intersect V intersect W, you have three ways to trivialize and you need a cocycle condition to hold relating the three different transtion functions. One of the standard ideas for how to build elliptic cohomology is to only require the cocycle condition to hold up to natural isomorphism rather than on the nose. I believe this requires you to replace vector spaces by an appropriate 2-category. My own idea was to use the 2-category of 2-vector spaces, which I learned about in a paper by Kapranov-Voevodsky. Is this idea worth anything?

  3. Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines. They wanted to use gerbes, or 2-gerbes maybe, instead. I could never understand what a gerbe was, but I am sure it is supposed to be a bundle of groupoids. I haven't heard anything about this for a while. Can this idea be made to go somewhere?

  4. Yet another idea is to go back to a decription of cobordism I once heard. I think this description is in print somewhere, but I don't know where or who wrote it. I believe this idea is a geometric description of MU^* X. Think of X as a simplicial set. Over each vertex of X put a manifold. Over each edge of X put a bordism between the manifolds at the vertices. Over each triangle, put a bordism between the bordisms on the edges--I don't really know what these means, but orientation must be involved. Continue in this way, and that is an element of MU^* X. There must be some equivalence relation you put on these, but I don't know what it is. The problem would then be to figure out whether this works and if it has been published, then to determine how you get K-cohomology from this description--presumably the bordisms between bordisms go away, so are the identity, and this is the cocycle condition for vector bundles. Then figure out how to get elliptic cohomology.

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Mark Hovey
Department of Mathematics
Wesleyan University