Response to: Tillmann, U. "Mumford's Conjecture—A Topological Outlook." In Handbook of Moduli: Volume III (Advanced Lectures in Mathematics). Vol. 26. International Press of Boston, 2013, pp. 399–429. ISBN: 9781571462596.
Dear Professor Miller,
Here is my response to the Tillmann Paper:
Well, the classes \kappa_i are very natural, and nobody could think of any other way to generate characteristic classes for surface bundles. That's when conjectures bubble up. What Mumford did not conjecture, because it requires topological techniques, is that the scanning map is an integral equivalence, not just rational.
The question of the relations among those classes in H^*(BDiff_g) is very interesting. I think the image is actually finite dimensional.
Also the spectrum MTSO(d) has been seen before. An alias is Th(-can | BSO(d)) where can is the canonical d-plane bundle and by Th I mean the Thom spectrum of the virtual vector bundle associated to -can.
Response to: Adams, J. F. "Quillen's Work on Formal Groups and Complex Cobordism." In Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics). University of Chicago Press, 1995. ISBN: 9780226005249. [reprint of the 1974 original]
Hi Professor Miller,
My response is below:
First of all, reading this paper was really satisfying! I've really only seen the usefulness of formal groups from the angle of number theory, so I enjoyed this a lot, and I'm glad I finally read it.
More particular things:
Lazard's theorem is equivalent to the statement that any (1d, commutative) formal group can be lifted across any surjection of rings. The annoying thing about Lazard's theorem is that it doesn't provide you with generators. This defect is one of the reasons for liking the condition of p-typicality.
Quillen's approach is to try to show that the generators of the MU formal group law generate MU_*. He does this using a version of Steenrod operations on MU_*.
Suppose you have a commutative monoid R in the stable homotopy category (a "ring spectrum") whose homotopy groups are zero in odd dimensions (eg K-theory). Then you can compute that R^*(CP^\infty) is a power series ring over R^*, and you get a formal group over R^*. The set of coordinates for this formal group are in bijection with the set of ring spectrum maps from MU.
Formal groups provide an interpretation genera for U-manfolds. All the parts of the theory of genera have interpretations in terms of formal groups. The characteristic series is x/exp(x), for example. I think Miscenko's theorem is exceptionally beautiful!