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.
- 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.
- 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?
- 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?
- 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.
Department of Mathematics