The strange adventure of the Universal Coefficient Theorem in the night

If you trawl the internet for more exotic Universal Coefficient Theorems, then you’ll come across comments to the effect that there isn’t a UCT for cohomology with local coefficients in general (for instance, here, for group cohomology, which is ordinary cohomology of a K(G,1), or the Groupprops wiki page, which only gives the trivial coefficients version). I had reason to want a UCT for group cohomology with coefficients a nontrivial module, to try to clear some hypotheses using formal properties of group homology (it’s better behaviour with respect to filtered colimits, in particular). The only reference I could find for a suitable UCT was an exercise in Spanier’s venerable book:

Page 283 of Spanier’s book Algebraic Topology

I found a reference in a 2018 paper, that said “there is a version [of the local coefficient UCT] in [Spanier], p. 283, though its application is limited”. Up until this point, I had not actually seen the statement written out anywhere! In particular, it’s not clear what Spanier’s assumptions are (it might be he is assuming R is a PID throughout this section, but I couldn’t see it on a quick search), and in particular, something must break for this to be of “limited” application.

Continue reading “The strange adventure of the Universal Coefficient Theorem in the night”

This argument of Mochizuki doesn’t make sense to me

I made this point in a comment at Not Even Wrong, but I think it worth amplifying. It is regarding section 2.3 of a recent document released by Mochizuki, which I reproduce below the fold, along with my discussion. I have taken a screenshot as I don’t know if the document will be updated or not. I’m trying to figure this out as I go, it’s just so bizarre [Edit as in, I was literally thinking it through as I wrote this, and I didn’t edit it, on purpose. It’s what David Butler tags as #trymathslive on Twitter].

Continue reading “This argument of Mochizuki doesn’t make sense to me”

Algebraic Topology: First Steps in Cohomology — videos available now!

tl;dr Here’s a free series of 24 lectures on algebraic topology!

In January and February this year, I taught an intensive course for the Australian Mathematical Sciences Institute (AMSI), as part of the annual summer school program. It usually rotates around the country, and students would normally all travel to the host university, and live and study together for four weeks. The course counts for a full semester’s credit towards their honours or masters coursework (I guess in the US system it would be roughly equivalent to a first semester grad course). However, this year things were … different, so the summer school was held online, though still hosted by the University of Adelaide, with lecturers (as usual) from around the country teaching (I ran my lectures on Twitch, inspired by Signum University).

Continue reading “Algebraic Topology: First Steps in Cohomology — videos available now!”

Algebraic topology class memes

Over the summer I taught a four-week intro graduate-level class on algebraic topology, with a focus on cohomology and homological algebra, using mostly combinatorial methods until the end, when we did singular cohomology, briefly. I encouraged my students to have a little fun in the class Discord server, and meme away. Here are some they come up with, plus one from me. For some reason, the little squiggly picture, which is meant to be the terminal 2-skeletal \Delta-set and that I called P, got the nickname ‘Pete’. The terminal \Delta-set, which I called P_\infty, got called “infinite Pete”. Some of these I don’t even know what they were in reaction to. But the skeleton one I know is because my frame rate was super super low before I got the OBS settings right, and the lag was … something else.

Daniel Kan got his PhD done in a real hurry

Daniel Kan was a massive influence in homotopy theory, algebraic topology and category theory.

“…Kan had to serve in the army. But the army allowed him to do his service at the Weizmann Institute itself, and he could thus stay there for another two and a half years. His job offered him a lot of spare time, and he began to think about topology again. In the Spring of 1954, Samuel Eilenberg came from Columbia University on a visit to the Hebrew University in Jerusalem. (…) Kan knocked Eilenberg’s hotel room door, and explained his simplicial description of homotopy groups. Eilenberg asked him if he could prove the homotopy addition theorem, and Kan returned a week later with a proof. Eilenberg told Kan that he had a thesis there, engineered an ad hoc arrangement giving Kan the status of graduate student at the Hebrew University, and in the summer of 1954 Kan submitted his thesis. He formally received his PhD in 1955.”

Daniel M. Kan (1927–2013), Clark Barwick, Michael Hopkins, Haynes Miller, Ieke Moerdijk

Kan’s thesis seems to have been published across four short notes (I,II,III,IV) in the Proc. Nat. Acad. Sci., all up consisting of 13 pages. The thesis itself was (as far as I can tell from the Hebrew University’s library record) only 37 pages. More details expanding on part IV later appeared in the Annals of Mathematics (that was 32 pages long).

Perhaps not surprisingly, this pattern continued, with his student David Rector writing a PhD thesis at MIT in 1966, titled An unstable Adams spectral sequence, that is only nine pages long:

Rector’s thesis comprises a title page, an abstract page, a table of contents page, 7 pages of math, a bibliography page (8 refs.), and a biographical note page. The MIT library record’s “9 leaves” exclude the title/abstract/contents, which are not numbered. Except for some trivial changes in wording in the intro, the mathematical part is indeed identical to the 4-page Topology paper, vol. 5 (1966), 343-346. The thesis occupies more space since it’s manually typed; not including section titles, the 4 sections are respectively 18, 23, 42, and 36 typewritten lines

Comment at MathOverflow question “What is the shortest Ph.D. thesis?“, Timothy Chow

The “4-page Topology paper” has as one of its pages the references (and whitespace), and the first page is a summary of the result (restated later) and a little background material. So the real content of Rector’s PhD thesis is contained in two pages!

Journal of Combinatorial Theory Series A editorial board quits; forms Diamond Open Access replacement

Good news for Open Access fans and non-fans of publishing giant Elsevier: another editorial board of an Elsevier-published journal (well, “most” of the editors) have quit and set up a replacement journal. The new journal is called Combinatorial Theory, and will be published starting the second quarter 2021 on the University of California’s eScholarship platform.

The new journal’s logo

Like a number of other good mathematics journals, it charges no publication fees and is free to read; the journal will publish articles under a Creative Commons Attribution (CC By 4.0) license. The Editorial Board of Comb. Theory is top-notch (including, I note, recent George Szekeres Medal winner Ole Warnaar).

Added, since Andrés Caicedo pointed out Ardila’s tweet thread on the matter:

Update on formal anafunctors

After some very helpful comments, I have managed to finish updating my paper on formal anafunctors (original release announcement post) and have now sent it back to the journal. The length increased by about 60%, as I had missed including the proof that the associator isomorphisms were natural (7 pages! Including the diagram in this post) and that middle-four interchange holds. The referee also pointed out that the covering maps don’t really form a subcanonical pretopology, but something a bit weaker, and this weaker notion is all I use. It wasn’t so much a matter of tweaking the definition, but recognising the weakness of the definition.

So here it is: The elementary construction of formal anafunctors, arXiv:1808.04552.

For amusement, two of the new diagrams…

Now I need to write the cospans paper

The implications!

“…p ⇒ q is actually equivalent to ¬(p ∧ ¬q), and undergraduate lecturers consistently get it wrong.” -Andrew J. Bromage (@deguerre)

😲 …. it makes so much more sense written like this! Why have I never been shown this before?? Link to thread: