# Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been worked on quite a bit since the time of writing. The list is very good as is, but there must also be a few good additions since 1999. Can someone point me to a more recent list of open problems in algebraic topology? My googling isn't turning up much else. Thank you!

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Although I have great respect for Mark Hovey, I have always felt this list to be very narrow in scope. For example, there is nothing about calculus of homotopy functors on it. Also, The list is quite silent about the study of the algebraic topology of manifolds. – John Klein Jul 4 '13 at 15:40
Hovey hasn't made additions, but in my time working with him I've found answers to several of these problems (some are in my thesis, others I found in published papers). At some point I may write an updated list which makes clear what's known (and where to find it), and what's still open. My focus was on the model categories, axiomatic stable, and equivariant lists. I won't have anything to say about the others. – David White Jul 4 '13 at 17:12
Hopefully he doesn't mind me advertising for him, but Tyler Lawson is presently assembling such a list which should be made public fairly soon. If you're anxious to see it, then you might want to email him about it, in case he doesn't see this question on his own. – Eric Peterson Jul 4 '13 at 17:59
He's Tyler Lawson. He sees everything – David White Jul 4 '13 at 18:52
Some of Tyler's remarks (and descriptions of beautiful, beautiful art) can be found here (notes written by Paul VanKoughnett at Talbot): math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf – Dylan Wilson Jul 5 '13 at 0:23

I have made a note of some problems in the area of Nonabelian algebraic topology and homological algebra in 1990, and in Chapter 16 of the book in the same area and advertised here, with free pdf, there is a note of 32 problems and questions in this area which had occurred to me. These problems may well seem "narrow", and/or "out-of-line" of current trends, but I thought the latter big book ought not to be seen as any kind of final product, but hopefully as the beginning of a new story. Also it is interesting to record for any area of study what it might be hoped to do but so far does not.

Further comment: The origin of this work was methodological (does that count as a "problem"?). In the 1960s, writing the first edition of the book which is now "Topology and Groupoids" (2006), I convinced myself that all of $1$-dimensional homotopy theory was better expressed in terms of groupoids rather than groups. So the next question was: are groupoids useful, or not, or to what extent, in higher dimensional homotopy theory? Trying to find answers to this has been a lot of fun for all concerned, though the published work with Philip Higgins was described once to me by Michael Barratt as "pursued in the teeth of opposition".

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Interesting! I cannot access the document now (I'm in the beach with a not so good mobile). Do you include among your problems the left properness of the model category of crossed modules? I know it's been open for quite a while. I thought it was just a technical point, but it showed up once in my work in connection with a computation in algebraic K-theory. – Fernando Muro Jul 5 '13 at 16:04
@Fernando: The problems listed are much more broad, but this question is mentioned on p. 594 (for crossed complexes- over groupoids) as Sauvageot claimed to prove it in his thesis, but there seemed to me to be a gap, concerned with calculating pushouts of crossed modules or complexes when there is a change of fundamental group. – Ronnie Brown Jul 5 '13 at 22:02

The website discussed in the comments went live a while back, and has a bunch of problems:

http://topology-octopus.herokuapp.com/problemsinhomotopytheory/show/HomePage

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