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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6$\begingroup$ 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. $\endgroup$– Eric PetersonCommented Jul 4, 2013 at 17:59
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9$\begingroup$ He's Tyler Lawson. He sees everything $\endgroup$– David WhiteCommented Jul 4, 2013 at 18:52
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4$\begingroup$ The ' problem' is that algebraic topology and even homotopy theory are such broad fields that any compendium is forcibly incomplete, unless it is done by many people from different places, and even different ages! $\endgroup$– Fernando MuroCommented Jul 4, 2013 at 20:19
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5$\begingroup$ 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 $\endgroup$– Dylan WilsonCommented Jul 5, 2013 at 0:23
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4$\begingroup$ Yes, there is currently such an effort (though it is not even remotely just an effort of mine). I would really hesitate to suggest when it might be public. $\endgroup$– Tyler LawsonCommented Jul 5, 2013 at 6:04
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2 Answers
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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 1981 published work with Philip Higgins was described once to me by Michael Barratt as "Carved out of solid rock, and pursued in the teeth of opposition".
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$\begingroup$ 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. $\endgroup$ Commented Jul 5, 2013 at 16:04
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$\begingroup$ @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. $\endgroup$ Commented Jul 5, 2013 at 22:02
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$\begingroup$ @Primegaphomologystonermon. I've replaced the broken links with snapshots on the Wayback Machine. An alternative to the second Wayback Machine link might be groupoids.org.uk/nonab-a-t.html. $\endgroup$ Commented Aug 10, 2022 at 22:41
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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
Update: It seems that this site is no longer accessible. Here is the last capture from the Internet Archive:
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$\begingroup$ The link has been down for several months now (at least since May). Does anyone have a copy? $\endgroup$ Commented Aug 10, 2022 at 9:25
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1$\begingroup$ @AivazianArshak Hopefully the Internet Archive link I added to the post has everything. All the links I tried were okay, at least. $\endgroup$ Commented Aug 10, 2022 at 21:15
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