12:32 AM
@MichaelAlbanese This is true at least when $m = 2$, and presumably at odd primes as well (although I can't find a reference off the top of my head). Note that any map $K(\mathbb{Z}, n) \rightarrow K(\mathbb{Z}/m, n)$ is a multiple of the one induced by the quotient map $\mathbb{Z} \rightarrow \mathbb{Z}/m$, so you're essentially asking if this quotient map induces a surjection $H^{\bullet}(K(\mathbb{Z}/m, n), \mathbb{Z}/m) \rightarrow H^{\bullet}(K(\mathbb{Z}, n), \mathbb{Z}/m)$.
This follows from the fact that the fundamental class is in the image of this map (in fact, under the Yoneda embedding it corresponds to the quotient map $K(\mathbb{Z}, n) \rightarrow K(\mathbb{Z}/m, n)$ itself) and that $H^{\bullet}(K(\mathbb{Z}, n), \mathbb{Z}/m)$ is generated by the fundamental class under products and Steenrod operations (see 1.3.7 in Hatcher's pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf at m=2)
12:52 AM
Dumb question: are there infinitely many even positive integers which are not divisible by p-1 for any prime p>=5?
Or in other words: does Z/24 appear infinitely many times as an image-of-j group?
I'm currently working on on editing the OEIS entry for "order of nth stable homotopy group of spheres", I guess cause somebody should. (It only goes to the 32 stem, and even so has errors.)

2 hours later…
3:05 AM
@PiotrPstrągowski: Thanks. I was aware of the $m = 2$ case (it served as partial evidence that my question wasn't a necessarily stupid one). I wouldn't be surprised if something similar holds for $m \neq 2$, but I wasn't sure. You seem to be of the opinion that it does. I guess one would need to know that $H^{\bullet}(K(\mathbb{Z}, n), \mathbb{Z}/m)$ is always generated by the fundamental class under products and cohomology operations.
3:35 AM
thank you, everyone, for your thoughtful responses and encouragement. i deeply appreciate it -- not just for myself, but because you are lending your voice to emphasize the importance of positivity and mutual supportiveness to our community, and of these sorts of reflective discussions as a necessary part of the process of achieving such an environment.
i have some more thoughts i will be sharing below. if you are not interested, please feel free to simply skip over what i say and proceed with your other discussions. at the same time, i recognize that it is not cost-free on the ambient environment for me to be posting these large blocks of text.
however, for reasons that i'll explain, i do think it is important and relevant to have this conversation here specifically. as i'll also explain, i expect not to be writing nearly so much as this here again anytime in the near future, so i hope this is alright.
@DylanWilson thanks for sharing that term, "mudita". it's wonderful that there's a word for that. i think we can all appreciate how important definitions are! they allow us to bring vague notions into our conscious thoughts. academia is definitely competitive, but the moment-to-moment experience is surely much richer and more joyful when one maintains an attitude of excitement and abundance, instead of one of scarcity and competition. all so very much easier said than done, of course.
regarding my sensations, i am glad you appreciated my description. i believe that all emotional responses are in fact carried in our body, and that in order to manage them and listen to what they are telling us, the first and most important task is to be aware of that connection. so, i shared my specific description towards normalizing this belief, in hopes that others might also notice the physical sensations that accompany their negative emotions.
@DenisNardin thank you for your kind response. as i said, i was already completely certain of your positive intentions, but i appreciate your apology nonetheless. in fact, in retrospect i am grateful that it happened, because it gave me an opportunity to reflect on the significance and value of this community to me.
this is brought into focus all the more by the pandemic; at least here in the US, it seems that we will not have any sort of in-person departmental community for quite a while, and i am realizing that this chatroom feels more like a mathematical home to me than any other space, real or virtual. i hope the utter seriousness with which i mean this is appreciated, towards justifying the length and scope of the thoughts i am sharing.
4
regarding bousfield--kan, it is Theorem 5.8 here: sciencedirect.com/science/article/pii/S0022404919300416
again, priority regarding that result is not such a big deal for me, but i nonetheless very much appreciate your reference request.
to be clear, i absolutely expect it to happen again and again -- to all of us -- that we are not always given the credit that we have earned. it would be impossible to avoid that. rather, what is important to me is that in such situations, we feel empowered to speak up for ourselves. as in all human endeavors, communication is everything.
@RizaHawkeye thanks for sharing your experience. it does feel vulnerable to share in this way. but as i uncover my own desires, i am realizing that most of all i am asking vulnerability of others, and so the best and only way to lead is with my own vulnerability.
you may well be correct that not many others here care. however, i would prefer to believe that it is primarily because this is simply a different tone of conversation than what is usually going on here. it may be that these messages do essentially nothing. that would be alright with me; all i can do is my best. but i would consider them to be a success if they end up being helpful to just one other person at some point down the line.
@S.carmeli thank you for chiming in! i am really glad you added your thoughts to the conversation. i have a few things i'd like to say in response, which really just lead into the bulk of what i wanted to say anyways.
first and foremost, i don't feel that i am particularly qualified to try to shape the community, either. i have long imagined that if and when i have a tenure-track job i will finally feel secure enough that i can begin trying to change things for the better -- that then and only then will i finally have the authority to do so.
in trying to decide whether to share my thoughts, i was acutely aware of the fact that i might come off as dictatorial or otherwise unsavory. ultimately, i decided that that was a risk i was willing to take -- that it was worth it to open this conversation.
in particular, i realized that power/standing/respect/etc. is a spectrum. (and it's not like you suddenly get a membership card to The Great Assembly Of Homotopy Theorists once you get a tenure-track job! i don't think...?) i have been around for a while now; i started grad school in 2010, and i've been an off-and-on participant here in the chatroom ever since it began in 2013. so, i can only hope that my speaking up paves the way for those who feel less established to do so as well.
in any case, i would like us all to operate under the axiom that our experiences are all equally important, regardless of where we stand. so, i hope that you and others feel empowered to share your opinions, if you feel compelled to do so.
second of all, i must tell you that i am extremely excited about your recent paper "ambidexterity and height", and yet i have not managed to actually take a look! but so, if you are up for it, i would be truly thrilled to hear you talk about it in here.
in this pandemic era of online seminars, i have been finding that while in theory i can always watch a seminar after-the-fact, in practice i cannot ever get myself to do it -- it is like pulling teeth. in the same way, i think that while in theory there may be little difference between reading your actual paper and hearing you talk about it here, in practice the latter will be both much more likely to happen and much more enjoyable.
in fact, i think i know the reason in both cases: i get the most out of mathematics when it is an interpersonal endeavor, and so i value interactive experiences far more than merely consuming mathematics in an isolated manner.
now, i absolutely agree with you that merely encouraging people to talk freely about their research is not an optimal way of distributing our attention, if the goal is to help everyone feel seen and heard. i was aware of this when i first wrote, but i didn't have the energy to go further down that route at the time.
indeed, in thinking more deeply about these questions, i realized that "a platform for people to talk freely about their research" is just scratching the surface of my own personal aspirations for this chatroom. in short, i would love to explicitly encourage discussion regarding all aspects of being a homotopy theorist.
i am not entirely sure what i mean by that, but i'll try to unpack it nonetheless. as a side note, i should mention that i feel inspired by eugenia cheng's "manifesto for inclusivity", particularly item 5: eugeniacheng.com/inclusivity
for people who are very early on in their career (say early grad students and undergrads (and below?)), one thing that's very helpful is to get a sense of the lay of the land: what directions people care about, what directions seem potentially fruitful but as-yet un(der)explored, how and where to learn, etc. etc.
i think we already do a decent job of fielding these sorts of questions, though of course there is always room for improvement. in any case, i would be happy for this to be codified as one of the explicit goals of this chatroom.
but a much bigger issue, in my mind, is highlighted by the fact that the world is utterly fucked right now, and that the sub-world of mathematics is very far from immune. in particular, the experience of "being a homotopy theorist" is extremely fraught, perhaps moreso than ever before. every stage of the trajectory from undergrad to tenure is its own tightrope walk. on a scale from 1 to 10, uncertainty is at like 15.
and this has real and lasting negative consequences, which are made many times worse by feeling alone. i would love for people to feel empowered to discuss their experiences, fears, hesitations, and so on. one of my collaborators needs to get a job this fall in order to stay in mathematics. he described his feelings about it as "like being in front of a firing squad, and all i can do is hope that they somehow manage to miss me".
wow. i feel so grateful that i happen to just be starting a postdoc this fall, and so am in as good of shape as i could hope for outside of having a tenure-track job; and yet i also relate so much to his feelings.
so to reiterate: there is going to be a lot of extremely difficult and stressful stuff that a lot of us are going to be having to go through (as if the usual job application process wasn't already bad enough!); and i would like for this to be a place where people can lay down some of their burdens, and share sympathy and empathy.
partly in the interest of explaining what i mean, i'll take another vulnerable step and share my own recent experiences. again, my hope in this is that everyone feels empowered to share their own experience, regardless of their current circumstances or anything else.
i am not aware of another avenue for such discussion; perhaps it exists, but in any case i think there is great value in having a single forum for this to occur (where we recognize each other's names/handles, and perhaps know some of the people IRL too).
this is because i believe very much in the definition of community as "a place that you can bring your entire self to". i quite like this essay that explores the fragmentation of modern culture and the corresponding absence of community in that broad sense (which is where i picked up that definition in the first place): ribbonfarm.com/2017/01/10/…
on the other hand, i'm completely open to the possibility that it would be better to have different channels for different types of conversation (such as on slack or discord). but that's a second-order question, which i'm not trying to address right now.
anways. when the pandemic first hit, i thought it would be great news for my research. summers have always brought up tension between focusing on research and traveling to conferences, and so i was perversely excited to have my hands tied. and for a while, it was great, although it took a while to find a balance. there was a week when i clocked 33 hours of zoom-math, between research meetings and seminars. that was way, way, way too much.
just over a week ago, i suddenly hit a block. it was on a thursday afternoon, and it was sudden, and it was a great shock. i have extremely deep reserves of willpower; i sometimes feel that this is my primary asset as a mathematician. but all of a sudden they ran completely dry. that had never happened to me before.
my guess is that pre-pandemic, there were always other parts of my life that would get in the way, before my willpower reserves ran dry; and by the time i came back to math, i would have built my reserves back up enough to continue.
but dry they were. so very dry. i am at the very tail end of a massive writing endeavor, but even with the end in sight -- close enough to taste -- i could not continue. it felt so awful. at times, i felt like i could barely breathe. this had never happened to me before. i once spent an entire month working towards a deadline and sleeping an average of 4 hours per night; and yet not even then did my reserves give out. but now they did. fuck this stupid pandemic.
and i thought: why am i even doing math. i am worthless. this is a worthless pursuit anyways. who am i even doing this for. i am wasting my life. i am wasting my one and only life.
so i stopped writing. i had no choice. i have never stopped like that before. i forced myself to sleep in, instead of waking up early to get more writing in amongst my meetings and other tasks. i spent a lot of time doing yoga, and playing piano, and meditating. just breathing; just trying to reassure my body that i am not, in fact, about to die.
and evidently, i've spent a lot of time thinking about what this chatroom -- and the mathematics community more broadly -- means to me.
i am writing this on sunday evening. i intend to return to writing tomorrow morning. (which is why i don't expect to be rambling nearly so much again anytime soon.)
but i feel scared. i feel scared that it won't have been enough -- that my willpower is still gone. that perhaps it will never return. and then who am i? it has been nearly half my life now that my single greatest goal has been to be a professional mathematician. it is not all i am, but it is so very much, and i have sacrificed all but endlessly for it. what will it all have been for?
on my best days, i know in my heart that it is the journey that is itself the blessing; but at the same time, the sense of impending failure weighs very heavily on me.
thank you for reading, and for giving me the space to lighten my load.

1 hour later…
5:14 AM
@AaronMazel-Gee Thanks for sharing this. Having had many similar experiences this year, I feel a lot of empathy with what you've written.
2
5:44 AM
@AaronMazel-Gee I completely understand much of the anxiety you express and the genuine desire to improve the community and make the world a more habitable place for the homotopy minded. You're great passion should be rewarded full stop. You've inspired me to chime in a bit too. Those of you who know me, know that I work in many different fields, but never shy away from learning new ones; never tell the mathematics where to go, it has its own mind.
I have the opportunity to share that this community is exceptionally welcoming of new people and tolerant of those trying to learn homotopy theory to further their own research endeavors. The problem of non-self promotion is pretty much universal among a specific class of mathematicians. Others, have no problem self promoting and wouldn't care if someone else did.
In terms of community, I can say I'd rather hear you make us aware of something you wrote rather than keep it to yourself to preserve some sense of humility. That all said, something I try to keep in mind is that math > time. Early in my career, I could work hard each day until "everything is as done as it can be". This idea I'm sure you're aware, but it gets worse the more you are exposed to. At some point, its important to realize you just have to put it down and get some rest.
There is more time for mathematics tomorrow. In terms of energy and willpower, those replenish with rest too. I won't stay on this meta topic too long, but I add my voice of support for the spirit of Aaron's comments, and absolutely feel this community has done great things to inspire, welcome, and support its members. I cannot say this about all other communities I've interacted with.
6:00 AM
@AaronMazel-Gee first, thanks for taking the time to talk about this and I'm sorry to hear what you're experiencing.
for the record, I've never had reserves like you describe. for years I've been managing periods of high activity followed by low; with responsibility changes the highs get shorter and the lows get longer. everything since grad school has essentially consisted of trying to manage this cycle.
the pandemic has completely shot apart my own coping mechanisms. I know we're not alone in having great difficulty getting things done.
5
I don't know how you work, but like lemiller suggests there is every reason to believe it'll replenish.
To make this not entirely "misery loves company", if you (or anyone else) want to commiserate, feel free to reach out to me privately at any point.
Take care, all.
6:59 AM
Thanks to everyone for bringing these sorts of things up. I was having a great deal of trouble getting much done after having a kid, to the point that I think my most two recent arxiv postings are verging on incoherent. And then the pandemic hit. Unlike Aaron I am extremely easily distracted, frequently unmotivated, and pretty much always tired.
Of course I can't really complain too much since I was lucky enough to get a tt job this past cycle, in a place that I like, but I just wanted to chime in and say that, like others, the pandemic and the isolation have been a really difficult atmosphere to be excited about math in.
But I will say that every time I log in to this room, I am heartened to realize that this room, which I created 7 (!?) years ago after an amazing workshop in Lake Tahoe (not far from where I'm living now!) with @EricPeterson and @AaronMazel-Gee and @DenisNardin and @TylerLawson and @DylanWilson and so many other amazing people, is still serving as a community gathering place.
3
But I think that is part of what's been so hard. The experience of sitting around a fire and talking about homotopy theory with brilliant friends until 3am, was one of the things I found so wonderful about being a mathematician. And I really miss it.
2
I would also like to say that I continue to be frustrated and saddened that this room has pretty much NEVER been a place that the young women in our field have felt comfortable. A friend once told me that any place where large groups of white men feel comfortable and at ease is probably a place toxic to women.
This has been a refreshing read and I don’t think I can say anything to improve on what has already been said here. But I have felt very isolated recently in that it seemed (from the other side of a zoom call of course) that everyone else in the community was doing just fine during this pandemic, and potentially being even more productive than usual. Starting this discussion has been helpful for me, and it will be helpful for many others, to hear that it is hard for everyone right now.
9
I'm not really interested in debating whether or not this is some kind of universal truth, as we've already been on that ride too many times to count, but I hope we can continue to be introspective about whether or not our behavior is welcoming and inclusive, bearing in mind that we may sometimes have to do something EXTRA to create such an atmosphere to compensate for the misbehavior of others.
In other words I think the moral and ethical course of action is not to say "Well, if others make MO/math/society toxic for certain people, all we can do is be nice and not make things worse." I think that, as thoughtful and able people, we need to actively work to undo the effects of oppression and inequity, not wash our hands of it.
5
I'm grateful to be a homotopy theorist, and a human being. Thanks both to the people in here who have been vulnerable and real, and to the people who help me stop being lost in the jungle of ∞-categories. I'm desperately trying to hack my way out.
Night night 😴
8:11 AM
@AaronMazel-Gee I have no idea if this is helpful, but I think of myself as pretty disciplined in regards to writing, and I'm also not doing as well as I wish in this regard currently. There are some projects which I have basically finished and which have been in a dormant ("just-needs-more-writing") state for a long time. I figured during the epidemic I would finally be able to catch up with everything. Not working out...
8:32 AM
@AaronMazel-Gee first, of course, I'll be happy to tell you about AmbiHeight on any platform any time :-) And I hope you feel better soon, I believe the feelings you express are very wide-spread in math and in general. Things will get better eventually (either by having a vaccine or by the academy adapting to the situation)!
8:50 AM
Just making sure I didn't write nonsense in some draft: If i start with the topological monoid of free C[G]-modules, but with the maps being non G-equivariant isomorphisms, and let G act on this monoid in the natural way (act on each free module by... you know... its a module...), then group complete to G-spectra. Do I get connective G-equivariant K-theory this way?
9:45 AM
@AaronMazel-Gee Thank you, Aaron, for sharing your thoughts and feelings. I hope your post will help contribute to lessening the burden of keeping up the facade that we are all on top of things (in professional/emotional/person/mathematical etc. matters).
9:59 AM
@AdrianClough So once we are in the land of spectra, things are indeed easy, as I can just assume that the lowest non-vanishing homotopy group is of degree zero, and then note that going from $\mathbb{S}$-modules to $\mathbb{Z}$-modules via $0$-truncation can be split up into first going to $H\mathbf{Z}$-modules, and then taking $0$-truncation. This leaves us with the Freudenthal suspension theorem, so I am wondering if the following version is true and has been written down:
- Consider the topos of parametrised spectra $\mathcal{PS}$, then for any space $X$, can I take its universal $\infty$-connected object? Let’s call it $\tau_{\geq \infty}X$.
- How does this related to $\Sigma^\infty X$?
- How do I work with pointed spaces inside $\mathcal{PS}$? Given a pointed space $X$ can I use $\tau_{\geq \infty}$ to obtain a map $X \to \Sigma^\infty X$ in $\mathcal{PS}$?
- Is this map $(2n-1)$-connected for $X$ $n$-connected?
I guess the last question should be: How do I make sense of the statement that this map is $(2n-1)$-connected for $X$ $n$-connected?
The above questions are motivated by the following: The more connected a space is, the closer it is to "being a spectrum". It would presumably be easier to make sense of this statement if spaces and spectra live in the same category, and $\mathcal{PS}$ seems like a natural candidate for such an ambient category.

3 hours later…
12:59 PM
@CharlesRezk robert burklund asks me to say: take 2p for p = 1 (mod 3).
@PiotrPstrągowski: I am considering asking the question on the site. Would you mind if I include a reference to your comments?
@MichaelAlbanese Sure, go ahead! It's an interesting question.
I'd also like to thank the others for being so open with their experiences, it's helpful to hear. part of what's been difficult about this pandemic is that now, more than ever, it's much easier to tie self-worth to productivity (e.g., the memes about newton working out gravity during the plague), motivation for which has dried up a fair amount for me.
2
@PiotrPstrągowski: Done, thanks.
1:43 PM
the discussion here has been heartening to read. while my "career" as "a mathematician" will be short-lived, i care deeply about the development of the field. but most of all, i care deeply about the experiences of the people who further homotopy theory.
math can be lonely -- indeed, this is one of the reasons i opted not to pursue a phd in math -- but the empathy and sympathy i've experienced from members of the homotopy theory community broadly (and the chatroom more specifically) have made my time learning this subject so worthwhile!
2
so, thanks y'all. i may not ever be actively doing research again, but i will continue following developments in the field with great interest, and hope to serve as an "emotional cheerleader", providing encouragement from the sidelines :)
@skd we should force him to do what needed to be able to come over and tell the solution himself :-)
I talk about Robert of course

1 hour later…
3:21 PM
Hi all, I have quite simple to dumb question. If I have a topological space X which homology w/ coeff's in a ring R is free, is it possible to write a map from a wedge of spheres to X that induce H_*R-isomorphism?
@IgorSikora No, the multiplicative structure on cohomology is an obstruction (because on a wedge of spheres every class in positive degree squares to 0)
Ok! I will try to find some counterexample
Thanks a lot!
3:38 PM
@IgorSikora $\mathbb{CP}^n$ is one :)
That exactly came to my mind - though I need a confirmation that a map between top. spaces that induces isomorphism in H_*R induces also isomorphism in cohomology and vice versa?
I.e being H_*R and H^*R-equivalence is the same thing?
@IgorSikora H_\ast R-equivalence induces an H^\ast R-equivalence by the universal coefficient theorem. I think you can get counterexamples for non-finite CW-complexes for the converse
Lovely, thanks!

6 hours later…
9:41 PM
quick question: where can I read what the map TMF -> KO does on homotopy groups? I'm most interested in what happens at 2. Thanks!
10:02 PM
I have a stupid question. Let G be a profinite group and consider the site BG of profinite sets with a continuous G-actions and where coverings are just continuous surjections. On the other hand, consider the site BG', where now again objects are profinite sets with a continuous G-action, but where covering of U is given by a family of jointly surjective maps U_i -> U such that each U_i \to U is open and continuous.
For a discrete G-module M, I get an object M \in Ab(BG) and M \in Ab(BG') by taking the associated sheaf, Hom(-,M). Now, there is an obvious map f:Ab(BG) -> Ab(BG'). Now, the claim is that R^i f_* M = 0 for i > 0. Why?
10:25 PM
@ArunDebray this is a ring map, so i'll say what happens on generators. unit maps to unit (obv), eta^n to eta^n, nothing hits 2v_1^2 (because pi_4 tmf = 0), c_4 goes to v_1^4, and 2c_6 goes to 2v_1^6. i think that's it.
@S.carmeli i think he doesn't have enough reputation to be able to chat in here
11:10 PM
@skd Your map sends Δ to 0, which is no good.
@ArunDebray Pardon my ignorance, but how is this map TMF -> KO constructed?
whoops, yes, thanks dexter. i was wrong earlier, i think c_6 maps to zero
there is a map tmf -> KO[[q]] given by q-expanding, and i think arun's considering the map tmf -> KO[[q]] -> KO. if what i wrote is correct, then Delta goes to v_1^something, and so this factors through tmf[Delta^-1] = TMF
Δ is a cusp form so it goes to 0 under this composition
oy vey, maybe i should shut up and let others give an answer
There is a map TMF -> KO((q)) that is given by taking q-expansion, and you are free to pick out the KO piece, but this is now not a ring map.
(well you can pick out any KO piece, and it will not be a ring map)
11:27 PM
yeah. i guess what i was trying to describe above is actually the map tmf_K(1) -> KO
i think that 2c_6 does map to 2v_1^4 under this map, but this may obviously be wrong
Yeah, skd's map is a map Tmf -> KO which corresponds to evaluation at the cusp, and sends Delta to 0 (so it doesn't factor through TMF). I don't think you expect any others except in the p-complete setting because the composite Tmf -> KU would be an elliptic cohomology theory whose formal group is the multiplicative group over Z.
In any case it is easy to compute the values of these maps at non-torsion points, because it's taking the constant term of the q-expansion. The target doesn't have lots of torsion so I'd imagine the torsion points are also easy to manage