Mark Hovey's Algebraic Topology Problem List

This list of problems is designed as a resource for algebraic topologists. The problems are not guaranteed to be good in any way--I just sat down and wrote them all in a couple of days. Some of them are no doubt out of reach, and some are probably even worse--uninteresting. I ask that anybody who gets anywhere on any of these problems, has some new problems to add, or has corrections to any of them, please keep me informed ( If I mention a name in a problem, it might be good to consult that person before working too hard on the problem.

Before proceeding onto the problems, I want to make a few polemical remarks about algebraic topology. The field is a small one, and to some extent we have been marginalized in mathematics. This is completely ridiculous, since the methods and ideas of algebraic topology have broad application to other areas of mathematics--witness Voevosdky's recent Fields Medal caliber work. We as algebraic topologists must bear part of the responsibility for this marginalization, and we must attempt to improve the situation. There are two ways we can do this. The most obvious method is to work on problems that arise externally to algebraic topology but for which the methods of algebraic topology may be helpful. This is a tough situation to get into--I don't think I have ever managed it--but very much worth it. Much of the action in mathematics in the last 10 years has come from interactions with physics, and algebraic topology can probably say more than it has. See any recent paper of Jack Morava for some ideas on this score.

However, even if the problems we work on are internal to algebraic topology, we must strive to express ourselves better. If we expect our papers to be accepted in mathematical journals with a wide audience, such as the Annals, JAMS, or the Inventiones, then we must make sure our introductions are readable by generic good mathematicians. I always think of the French, myself--I want Serre to be able to understand what my paper is about. Another idea is to think of your advisor's advisor, who was probably trained 40 or 50 years ago. Make sure your advisor's advisor can understand your introduction. Another point of view comes from Mike Hopkins, who told me that we must tell a story in the introduction. Don't jump right into the middle of it with "Let E be an E-infinity ring spectrum". That does not help our field.

Here are the problems:

Other topics that might be good to make problems for: Goodwillie-Weiss calculus of functors,

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Mark Hovey
Department of Mathematics
Wesleyan University