From: Dev Sinha Subject: Re: Elliptic cohomology Date: Wed, 17 May 2000 14:07:03 -0400 Newsgroups: sci.math.research Summary: [missing] Hi John, I'm not an expert in this subject, and I won't fully answer your question, but I think I have a couple things to say which might help. First is to think of the Index Theorem ("the case down the chromatic ladder from elliptic cohom"), which applies to any smooth, compact manifold. While the theorem applies to any compact mfld with an elliptic operator, the cohomology theory which plays a vital role is K-theory, which is complex-oriented. The topological index is best defined via K-theory. The trick is that for any X, TX is naturally a complex manifold. A second thing to note is that sometimes complex-oriented theories have "companions" which are not complex-oriented. The prototypical example is real K-theory, which is not complex-oriented but can be built from K-theory if one pays attention to the Z/2 action coming from conjugation. The versions of elliptic cohom constructed by Hopkins and Miller, which are now being called rings of topological modular forms and one of which is the proper target for the Witten genus, follow this pattern. In a fairly unrelated matter (of which I am reminded because we are talking about orientations), I'd like to share a bit of knowledge which I learned embarrassingly recently: We all know that in order for a manifold to satisfy Poincare duality for integral coefficients, it must be oriented. What about PD for other cohomology theories - what conditions are needed? Algebraic topologists know well that what you need are Thom isomorphisms for the normal bundle of the manifold in question, but can one get more specific? Might usual orientability be enough? When the cohomology theory in question is real K-theory, there is a beautiful answer: there is PD for KO-theory if and only if the manifold in question is spin. According to a friend who studies elliptic operators on manifolds, the KO-theory orientation class is, in an appropriate setting, represented by the Dirac operator. - I know this is old hat to some, but I was happy to learn it recently. -Dev On 14 May 2000, John Baez wrote: > Some people, when talking about elliptic cohomology, make a big > deal of the fact that it's a kind of quotient of *complex* cobordism > theory: they talk about how the complex cobordism ring is the Lazard > ring, associated to the universal formal group law, and then talk > about how other formal group laws, like the elliptic one, give > other complex oriented cohomology theories. So why does Charles > Thomas, in his book "Elliptic Cohomology", start out putting such > emphasis on *oriented* cobordism theory? In particular, he's really > interested in how the oriented cobordism ring, tensored with Q, is > a free polynomial ring with generators corresponding to the spaces > CP^{2n}... while someone like Ravenel, in his book on "Complex Cobordism > and Stable Homotopy Groups of Spheres", is very interested in how > the complex cobordism ring, tensored with Q, is a free polynomial > ring with generators corresponding to the spaces CP^n. > > Another way to put this is: what do elliptic genera have to do with > elliptic cohomology? The former are defined in terms of oriented > cobordism theory, while the latter seems to be more closely related > to complex cobordism theory [via formal group laws]. > > What am I missing? ============================================================================== From: baez@math.ucr.edu (John Baez) Subject: Re: Elliptic cohomology Date: 18 May 2000 23:42:35 GMT Newsgroups: sci.math.research Summary: [missing] In article , Dev Sinha wrote: >I'm not an expert in this subject, and I won't fully answer your question, >but I think I have a couple things to say which might help. Your comments were very helpful, as were those of Mark Hovey, who assured me that after we invert the prime 2, oriented cobordism theory and spin cobordism theory become the same, while complex cobordism becomes *almost* the same. More precisely, we've got a spectrum MU for complex cobordism theory, and we've got a spectrum MSO for oriented cobordism, but localizing away from 2 MU becomes isomorphic to MSO wedged with the double suspension of MSO. What this means is that lots of stuff can be done either for oriented cobordism, spin cobordism, or complex cobordism, without it making much difference - as long as we don't care about 2-torsion. Apparently the authors who were confusing the heck out of me by using oriented cobordism to study elliptic cohomology were actually trying to be nice, they were afraid I'd be scared of complex cobordism! (For years I had an instinctive fight-flight reaction whenever anyone did something algebraic like "localization" to a spectrum, so I sympathize with anyone who feels that way, but I think that in the last couple of months I've overcome that instinct, so now I'm gonna talk like I've understood this stuff since birth.)