# Applications

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

1. Introduce stable homotopy theory into the world of C^*-algebras, like Voevodsky has done in algebraic geometry. More specifically, find a model structure on some category of C^*-algebras that is useful. Then stabilize it somehow to find a stable homotopy category of C^*-algebras where Kasparov's KK theory is just Hom in the category. This will probably be the same as finding a category where K-theory of C^*-algebras is representable. I have thought about this a little--almost the first thing you realize is that C^*-algebras are not closed under inverse limits, so you need pro-C^*-algebras. These have been considered by Chris Phillips in N. C. Phillips, Inverse limits of $C\sp *$-algebras, J. Operator Theory {\bf 19} (1988), no.~1, 159--195; MR 90c:46090. Another thing you realize is that it is easier to map into C^*-algebras than it is to map out of them, and so it seems likely that any model category will be fibrantly generated instead of cofibrantly generated. But not much else is clear to me. One of the people who knows the most about the combination of C^*-algebras and homotopy theory is Claude Schochet. The viewpoint I am advocating here is shared by Jim McClure, and probably Marius Dadarlat, and they may know more about this. In particular, C^*-algebraists often make an assumption about their cohomology theories which basically forces them to be K-theory--I think it is Bott periodicity. To do what I am saying, you would have to drop this assumption.

2. Neil Strickland points out that several different moduli spaces are used in differential geometry and physics. For example, there is the moduli space people are always talking about in every talk on gauge theory I have ever been to. I can't remember what this moduli space is now, unfortunately--something to do with connections and the gauge group? Anyway, the geometers tend to think only about the rational homology of these spaces. What about their Morava K-theory? (Neil also mentioned solitons and Seiberg-Witten theory).

3. Investigate Voevodsky's stable homotopy category of schemes from a homotopy theorist's point of view. This is obviously a huge, unstructured problem, but I think there is room for a serious algebraic topologist to make some inroads. In particular, Bousfield localization is really a great thing, and they have not exploited this at all so far, I believe. The drawback is having to learn so much algebraic geometry before you have a chance.

4. Stefan Stolz showed that a simply connected Spin manifold of dimension at least 5 admits a metric of positive scalar curvature if and only if its image under the orientation MSpin --> KO is 0. But the situation is not completely understand when the manifolds are not simply connected. It seems to get involved with the Novikov conjecture. Oh yes, I remember--it is actually false for some fundamental groups, but the trouble appears to be all with products with the Bott manifold. That is, the general conjecture is: suppose M is a Spin manfold of dimension at least 5. Then a product of M with some finite number of Bott manifolds admits a metric of positive scalar curvature.

5. Try to carry out Stolz's plan for metrics of positive Ricci curvature. Here we expect the obstruction to lie in elliptic cohomology rather than K-theory, and the manifolds should be MO8 manifolds. But hardly anything is known here, I think.

6. Improve on Benson-Carlson-Rickard. Recall their theorem: if G is a finite p-group and k is an algebraically closed field, then thick subcategories in the stable k[G]-module category (= category obtained by killing projectives = injectives = frees) are in 1-1 correspondence with subsets of Proj H^*(G,k) closed under specialization. There are several open questions related to this. The most obvious one is to remove the algebraically closed requirement. This may involve understanding some Galois theory of stable homotopy categories, but may not be too difficult. I sincerely hope that anybody who wants to work on the stable module category will make the effort to understand my work with Palmieri and Strickland on this; our setup makes things considerably simpler, in my opinion.

7. Classify the localizing subcategories of the stable k[G]-module category. These should be in 1-1 correspondence with arbitrary subsets of Proj H^*(G,k). I have no idea how to do this one.

8. Extend the results of Benson-Carlson-Rickard to connected, cocommutative Hopf algebras over a field, like A(n). Hovey-Palmieri have achieved some partial success here; we reduce the calculation of thick subcategories to the case when the Hopf algebra is quasi-elementary. For k[G], quasi-elementary subHopf algebras look like k[E] for elementary abelian p-groups E, and the group cohomology is a polynomial algebra, so Benson-Carlson-Rickard use algebraic geometry to finish the proof (this is where they require the field to be algebraically closed). For A(n), the quasi-elementary subHopf algebras are also exterior, but the Ext groups are bigraded. We don't know how to do bigraded algebraic geometry, so we are stuck there. But I think something should be doable here. Anyway, here is the abstract of the Hovey-Palmieri paper , and here is the dvi file.

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Mark Hovey
Department of Mathematics
Wesleyan University
mhovey@wesleyan.edu