in BCSS leads to a contradiction, for a suitable compact connected Lie group, taken to be for concreteness. Recall that I had calculated the difference between a pair of connections on the -bundle , one arising from the construction of Murray and Stevenson, and the other being the pullback of this along the lift of for a path in starting at the identity matrix. This lift is defined in BCSS using . In actual fact, we did the calculation on the trivial bundle over that is the pullback of the one I care about, and arrived at the formula

where evaluates a path at the endpoint. I can show that for a careful choice of , path and tangent vector at that is vertical for , the right hand side fails to vanish. The left hand side is identically zero for vertical tangent vectors, and this will be the contradiction.

To start, we need to calculate when evaluated on a pair of tangent vectors. Since is left invariant, this simplifies matters, so that we have

where since I merely need to show this is nonzero, I ignore all prefactors in the calculation here and below. But taking the definition of , this gives

But, however, I have to pull this back along the evaluation map , integrate this over the fibre of the projection , and evaluate it on a tangent vector . This gives the following:

Now at this point we need to pick some actual paths and so on. Since we are in , I can write things out in the basis for the Lie algebra given by the (anti-Hermitian version of the) Pauli matrices, namely

Now a vertical tangent vector will be a loop based at the constant loop at , so let us take

For the path , based at the identity matrix , let us take

.

And for the path , let us take

From these definitions we get and . Finally, we can plug these in to our integral and get

Using the commutation relations for the Pauli matrices , the trace term becomes , and since we are merely looking to find something nonzero, we are left with the integral

which is manifestly positive. We thus find that for our choice of -vertical tangent vector and path , , contradicting the fact .

So what is the fix? Basically, the issue arose because for the given definition of , BCSS get the identity . If instead we had , then in the calculation in the previous post, Stokes’ theorem for integrating over the fibre would instead give us . Making the definition

avoids the contradiction. In follows that in the statement of Proposition 3.1 of BCSS, the resulting crossed module of Lie algebras has the action

where I’ve used integration by parts to rearrange slightly. This means that there will have to be some sign changes elsewhere in the article. This is constrained by having to get the definition of (weak) morphism of 2-term -algebras to agree with the definition of Lada and Markl, and to make sure the theorem of HDA VI relating the 2-cateories of Lie 2-algebras and 2-term -algebras still holds. I am confident this can be done, since in fact only the special case relating crossed modules of Lie algebras and 2-term dg-Lie algebras needs to applied here, leaving some extra flexibility in how the definitions for Lie 2-algebras need to be arranged, if at all, to ensure the equivalence of 2-categories exists.

]]>in the proof of Proposition 3.1 cannot be correct, since it leads to a contradiction in a geometric calculation independent of all the -algebra material. I should note that this geometric calculation is nothing exotic; a correspondent let me know I muddied the water in the previous post by discussing it in the context of a larger project with more novel constructions. As a result, I will give the calculation later below the fold.

What I did not realise when I wrote the first post is that the sign error in is linked not only the an overall sign in equation (5) of BCCS, but also to a plus sign implicit in the definition of the comparison functor from 2-term -algebras to (semistrict) Lie 2-algebras, which are possibly weak Lie algebra objects in the 2-category of groupoids internal to . This functor is constructed in HDA VI, in Theorem 4.3.6 (=Theorem 36 in the arXiv version), and here is the relevant version

This was invisible to me, until I checked a different article generalising HDA VI, namely Roytenberg’s *On weak Lie 2-algebras* (or free arXiv version), where a generalisation of the comparison functor to weaker objects is given. Here is included a minus sign on what turns out to be the troublesome piece of data arising from the crossed module of Lie algebras, arising from the sign on .

So now in fact one can see the sign on as being a combination of *two* signs: one from the comparison functor, and one from the coherence equation (5), just that in BCSS, the first of these is a , and the second is a .

Now the interesting thing is that Roytenberg uses a version of the comparison functor where there is explicitly a minus sign built in, where HDA VI has a plus, but also uses a version of the coherence equation that reduces to (5) in BCSS, once the additional assumptions of BCSS are applied. So we find ourselves in the position of having three papers, none of which agree about two different signs:

coherence equation | comparison functor | |

HDA VI | ||

BCSS | ||

Roytenberg |

To get a consistent version of Theorem 5.1 together with Proposition 3.1, given a choice of convention, the sign out the front of must be the product of the signs in the given row. BCSS get a minus sign, whereas if one used either of the other two conventions, it would be a plus sign. As we will see below, the minus sign in front of leads to a contradiction. Urs Schreiber’s point is that the sign in the first column should be dictated by the definition of morphism of -algebra, so this is going to force the sign in the comparison functor. Before doing this, however, I want to lay out my own computation, for why I think the sign on is incorrect. This will just be the first part, as it takes some background in geometry to set up. I will conclude with the actual contradiction in the next post, before trying to tackle the -algebra material.

The construction in BCSS takes a generic simply-connected, simple compact Lie group , but we can just take . Now is the infinite-dimensional Lie group of smooth functions such that . Then there is a (closed, Lie) subgroup satisfy additionally $late \gamma(2\pi)=I$, making the kernel of the smooth homomorphism . In both of these the group structure is pointwise, and we can identify tangent vectors with left-invariant vector fields, and hence with elements of the respective Lie algebras, which are and , of paths and loops based at the origin. We also need to use the group , which can be considered as functions such that , and . There is a principal -bundle that is also a central extension of Lie groups, with a connection defined in various places, for instance by Murray and Stevenson.

Now the geometry in question is that is a normal subgroup in , and one can lift the conjugation action to the central extension , as shown in Proposition 3.1 of BCSS. This is where the 1-form comes in. The definition of both the central extension and the connection on it go via working up on and then by showing everything descends along . Since is contractible, the principal bundle has a section, and the central extension even becomes a semidirect product of Lie groups, namely . The connection up on I will denote as , as is defined to be

where: is the evaluation map, is integration over the fibre, and is a left-invariant 2-form given by

.

Recall that are loops based at the identity. It should be noted that the definition of integration over the fibre of Murray–Stevenson uses contraction of a lifted vector in the *first* slot of the 2-form, not the *last* slot, as is sometimes the case.

The lift of the function to is defined first as a map , and then shown to descend. This map is given by

although in BCSS it is written in more elementary terms. At this point it is not important what is, all we need to know is that BCSS establishes that on (I have checked this myself, it is a direct computation, as stated in BCSS). I shall write for the restriction of to and similarly for .

Where the contradiction arises is when one tries to establish the analogue of this last identity, linking and , at the level of connections. It is true on general principles that descends to a 1-form on , but the question is, what is this 1-form? We can more easily answer the question for the lifted connection , since we have explicit formulas for it and for . The answer should be a 1-form on that descends along .

We can calculate as follows:

where in the second-to-last line I have used Stokes theorem for integration over the fibre, and the fact is the zero map on forms (be aware though that with the definition of from Murray–Stevenson, this means that the sign is different to the version of Stokes at, for example, this math.SE answer). Rearranging this we get

The left side vanishes when evaluated on tangent vectors vertical for , so to arrive at a contradiction I can evaluate the right side at a specific point on specific vertical vectors and get something nonzero. Despite the form of the identity, it isn’t immediately obvious this will happen, so I will plug in some actual paths and matrices next time to check.

]]>I like the way this type of problem can actually open up a dialogue of how mathematics is really open ended, creative and unfinished. Framing a fruit equation problem as a filter that makes people either feel stupid for not understanding it (it is basically algebra, after all), or smug and superior for getting it, is extremely unhelpful. But explaining that what can seem like an easy problem actually requires a lot of work, and would stump professional mathematicians (cf Fermat’s last theorem), is a good conversation starter. It can also touch on things like how geometry is brought to bear on algebraic problems (and vice versa!), how the solution can use methods not present in the problem and so on.

So, in honour of this, I thought I take what seems to be, at time of writing, the simplest currently unsolved Diophantine equation (that is: a multivariable polynomial, and looking only for whole-number solutions), and turn it into a fruit equation. We can think of it as trying to count fruit:

Here ‘simplest’ is according to the notion of “size” defined in this MathOverflow question, basically it’s a measure of how large all the powers and coefficients of a multivariable polynomial is. There are only finitely many polynomials of a given size. The polynomial from the picture is , and has size 29. Every polynomial of size 28 and smaller has either been solved or shown to have no solutions. The idea is to see (experimentally) where the rough threshold is between equations than be solved in a more-or-less elementary way, and where really serious techniques or obstructions really kick in, of the sort outlined here.

I’m happy for people to share the above image as widely as they can. If nothing else, maybe someone will actually solve the equation above, and contribute a piece of new knowledge!

]]>I have been very slowly working, with my colleague Raymond Vozzo, on a project that involves a very explicit calculation for bundle gerbes, and in particular with the gerbe underlying the crossed module of Lie groups introduced by Baez, Crans, Schreiber and Stevenson (BCSS), which presents the *String 2-group* of a suitable compact simply-connected Lie group G. However, there was a particular step where we got thoroughly stuck, where a certain calculation involving the comparison of two explicitly described connections on (a pullback of) the universal central extension of Fréchet Lie groups.

Aside from needing to overcome a sign convention around the Stokes theorem for integration along a fibre of a family of manifolds with boundary, there was still an issue where I got a formula that implied a certain non-zero 2-form was equal to its own negative. The origin of this negative sign is, ultimately, the minus sign in the definition of a certain family of 1-forms on appearing in the proof of Proposition 3.1 in BCSS. In my own notation it is

where is a smooth path starting at the identity element, is the left-invariant Maurer–Cartan form on and is a suitably normalised version of the (pointwise-evaluated) Killing form on . So I really couldn’t figure out what was going wrong, since I was fairly confident in our computations. This computation fed into a much larger computation, where if I ignored this problem and pressed on, trusting my instincts, things seemed to be going smoothly, and reproducing something like what we’d expect. I also had another (slow-burn) project involving this same comparison of connections, where the sign issue was also arising. The theory there pointed in one direction, the computation in another.

So much for the formulas. The whole point of the proof of Proposition 3.1 of BCSS was to prove that not only is a crossed module of Lie groups (the forms are used to define the action of on ), but also the induced crossed module of Lie algebras is the same as one defined earlier in the paper, denoted . More generally, there is a version of all of this with the “level-” central extension , and a corresponding Lie algebra crossed module . Now crossed modules of Lie algebras give rise to 2-term algebras with trivial ternary brackets (i.e. associators).

The import of this crossed module is proved later in the article, in that it is weakly equivalent to a certain canonical -algebra deformation of the trivial crossed module at integer levels. This was defined by Baez and Crans in HDA VI, and involves the trilinear form defined on . The proof of this equivalence goes via establishing a split short exact sequence of 2-term -algebras

where is contractible (for a certain notion of “contractible” for -algebras). The splitting is, importantly, an *up-to-homotopy inverse* of .

Now comes the funky bit. I had good reason to believe based on my second project that the 1-forms would “work better” if I just replaced them by . So I wondered if I could just make a new definition of the crossed module from BCSS and maybe prove it isomorphic to the one given there. No such luck, but then I wondered what would happen to the rest of the paper’s results, namely the comparison at the -algebra level. The important part is that this String 2-group is classified by a generator of a certain cohomology group, and if I made a new definition, I had better get the classification right, and that was reflected by the behaviour at the algebra level.

So if I flipped the sign on , then the crossed module of Lie algebras had a sign flip in the definition of its action. Then I’d need to check the short exact sequence given above. Now a morphism of algebras carries with it some coherence data relating the 2-ary brackets of its source and target, and one of these 2-ary brackets in the case of a crossed module arises from the action just mentioned. So if one just flips the sign on the appropriate bracket on one -algebra, then one might need to also flips sign on the bracket of the one at the other end of a morphism, and then the coherence data of the morphism definitely must also be tweaked.

Going back to the short exact sequence, the appropriate piece of coherence data for has a slightly out-of-place minus sign, so I could flip that no problems. In any case, is trivial (the relevant 2-ary bracket is zero so a sign flip does nothing) and somewhat subsidiary to the main proof. But the -algebra is set in stone, its brackets cannot be altered. The coherence data for could be tweaked with that extra sign flip, and things would be ok…for now.

But now I don’t just have a short exact sequence of -algebras, but a *split* short exact sequence: there’s a section that needs to be checked. If I follow through the consequences of of flipping the sign on through all the definitions, then at exactly this point I find *wouldn’t* be a morphism of -algebras. Everything else so far works by flipping signs judiciously, but the verification of a single coherence condition for breaks down. This condition is equation (5), in Definition 2.7 of BCSS:

This is a slight headache (and one has to replace all the s by s for everything that follows), but when I compare it to equations (3) and (4), namely

then I notice that the right hand side of these two equations looks like , whereas the left hand side of (5) looks like . This is not much to grasp hold of, but it looks slightly odd that the equations don’t quite follow a pattern, in which the term has a positive sign. Of course, a bunch of these functions are antisymmetric in their inputs, so perhaps some slight rearrangement could be done. But all the inputs in all three equations are in alphabetical order, which seems rather canonical.

Just on a lark, I thought I’d look at the paper HDA VI, since that is where all this stuff actually comes from. Then it turns out the definition of that third coherence condition, equation (5), for a morphism of 2-term -algebras is written completely differently:

The trivial notation shift can be ignored, what you should notice is that the terms can be rearranged to give the form as in (5) from BCSS, but then then the rest of the terms all collect to the other side, and we need to make use of antisymmetric properties to get all the arguments in alphabetical order.

And here, finally I strike gold. If I rearrange the third coherence condition from HDA VI, and get all the non- terms collected together and with the arguments in the right order… I get minus what is given in BCSS! The upshot is that the whole chain of definitions running downstream from are all dependent on getting the map of -algebras to be coherence, except as defined it *isn’t coherent using the definition in HDA VI*, which is the source text for this material. It’s also from Alissa Crans’ PhD thesis, where I trust that she did the necessary legwork appropriately, checking these nasty coherence conditions and their relations to other definitions of related objects.

Turning this argument around, to make coherent, we see that there is a chain of sign flips, namely of:

- in Lemma 5.7, leading to
- in Lemma 5.5, leading to
- in Lemma 5.4, leading to
- in Proposition 4.2 ,

leading all the way back to a sign flip in the proof Proposition 3.1. The paper as a whole is perfectly fine, as far as I can tell, once one makes this list of sign flips and fixes equation (5) by swapping the terms on the left hand side. This then makes our calculations come out consistent (more on these soon, I hope), and also makes my conceptual re-casting of the BCSS model construction work out.

Of the many people who have cited BCSS—54 are currently recorded by MathSciNet (subscription required)—I don’t think that many of them have actually needed the explicit definition of the action of the crossed module as we have, if any. I let the authors know, and John Baez took time out of his busy schedule to double check that there was indeed a sign error in (5), so I’m fairly confident I’m not making an error here. I also got John’s ok to go public with this error, since it’s not crucial and I can see how to fix everything.

]]>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 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.

So I asked on math.stackexchange, and Oscar Randal-Williams gave a great answer, pulling out something from the even more venerable source of Cartan and Eilenberg’s *Homological Algebra*. So I will copy the statement here (under the CC-By-SA 4.0 license)

If is left hereditary (eg a PID) and either is an injective -module (unlikely) or else is a complex of projective -modules (which holds iff the are projective -modules), then page 114 of Cartan-Eilenberg gives a standard-looking UCT, of the form

Oscar Randal-Williams, math.stackexchange answer

The case that I was looking at was , working over , so it is a PID, and my local system has each (technically, it’s for a connected Lie group), a free and hence projective module. Since the s are finitely generated free abelian groups, they are reflexive, and so taking in the above, I can switch the position of the dual from cohomology to the outer terms (this relies on knowing the standard double dual map is natural, so the whole dual local system is what is being swapped here, not just something on fibres).

What I was interested in is knowing when my cohomology group—of a space of the form where is a discrete group—was torsion, based on the group-theoretic properties of . Having a hypothesis involving cohomology using arbitrary local systems, even if of a special restricted form, was a bit too generic for my liking. However, with the above UCT, I can reduce the hypothesis on for arbitrary local coefficient systems with fibre , to one on . This might not seem like an improvement, but I’m looking as a special case at groups that are *locally finite*: the filtered colimit of their finite subgroups, and this means I can say something meaningful about torsion in the homology of , and hence in the cohomology.

My thanks to Maxime Ramzi for discussing this and related matters with me via Twitter chat, helping me past all kinds of silly misunderstandings, and to Oscar Randal-Williams for his answer on stackexchange.

]]>Here’s mine:

I think it captures my work pretty well. I do like bundles of groupoids internal to spaces, after all.

]]>Here is the relevant section. My discussion below will be performed using standard notation and terminology, which clearly doesn’t affect my conclusions, and makes things easier to read.

The argument in (ii) seems to me to be the following: if we work in a skeleton of the category of topological spaces, where all one-element topological spaces are actually equal (because there is only one) then the interval is collapsed to the circle . But doesn’t this argument also imply that every topological space is collapsed to a single point? Because, given a topological space with at least two points , we just run the argument in (ii) with the obvious substitutions, so that and are “identified” (I will explain the quotation marks below). But and are arbitrary, so that all pairs of points are “identified”, and so is replaced by the (unique) single element space, the quotient by the resulting equivalence relation. As a result, *every non-empty topological space is isomorphic to every other one, *in particular to the one-element space. This would seem to imply that the skeleton of the category of topological spaces collapses to the interval category with two objects, and one non-identity arrow.

The only way I can try to resolve this is that there is a purely linguistic sleight of hand here. Clearly working in a skeleton of the category of topological spaces in no way forces all non-empty topological spaces to collapse. But the use of the word “identified” here is ambiguous, and has two meanings in the current discussion:

- We can say in linear algebra that we identify any real -dimensional vector space with , meaning that up to isomorphism, we can assume we are working with the latter. There may be some minor details that need care, like whether the canonical basis of is being implicitly used or not, compared to a generic vector space without a chosen basis. This is mathematical, and well-known.
- But we can
*also*say that given a topological space, we identify a pair of points in that space, meaning that we perform a quotient construction. This changes the space we are working with completely, resulting with something not isomorphic to the original (this is the point of 2.3.ii), ignoring the issue of skeleta).

But the verb “identify” here is overloaded, and doing different things in each situation. This is obvious to any mathematician. And the complaint of Mochizuki is that his critics, what he terms “the RCS” (namely, Peter Scholze and Jakob Stix, I don’t know why he doesn’t do them the courtesy of naming them), are doing the former, but he is pointing to the latter as giving rise to some kind of problem. But this metaphor is *at best* a metaphor: it is not “closely technically related” to IUT, since i) and ii) are dealing with different meanings of the word, applying in different situations. The indented sentence starting “it is by no means…” is completely irrelevant, and I feel confused, since the sense in which and are identified as being isomorphic in the first half of the sentence is completely different from the sense in which they are identified as being equal points in a quotient space in the second sentence.

If Mochizuki’s writing was less circumlocutory (why do we need fancy notation for basic objects? and reminders of how they are built?), then it would be easier to grasp these apparently illustrative and enlightening examples. But it’s taken me several readings to understand that this seems to be a metaphor, at best, rather than a blatantly false statement. But the metaphor is being conflated with the situation it is meant to be explaining, due to a quirk of terminology. To quote the man himself from section 2.2 of the document “In fact, of course, such “pseudo-mathematical reasoning” is itself **fundamentally flawed**. [sic]”. I don’t find the pseudo-explanations involving line segments as metaphors for morphisms in a category at all useful, and indeed I find them actively misleading for people who aren’t familiar enough with category theory to take the metaphor of 2. above as being a faithful representation of the situation in 1.

**Added:** There was some discussion on Reddit about this post. I made some responses around points that I realised I failed I made clearly enough, and now I think my issue with the above example is that it is a total straw man. Part i) could have been written without reference to or the specific subspaces and , and it’s really just talking about terminal objects in a category, and the possibility that one can assume there is a unique terminal object, as long as one is willing to replace the category one is working with an equivalent one (you don’t even need to pass to a skeleton). Reusing the notation and terminology from i) in ii) seems chosen to conflate the two in the mind of the casual reader (it did for me, and I know better!). The indented sentence could be phrased more accurately and transparently as

“the fact all terminal objects in a category are uniquely isomorphic to each other doesn’t imply taking a coequaliser won’t give you a new object”

which is true, but tells us nothing about the veracity of Scholze and Stix’s argument.

]]>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).

I was flattered to be asked to put in a proposal for a course, and I thought I’d brush off my notes from the last time I taught algebraic topology, and focus more on cohomology, which I felt had to be covered in less detail than I would have liked (more focus was put on covering spaces and fundamental groupoids in that outing). The idea was to work combinatorially as much as possible, using -sets, and talk about topological spaces only to the extent that is needed to define singular cohomology. I also wanted to air an original proof of the long exact sequence (aka zig-zag lemma aka algebraic Mayer–Vietoris) that I had learned from my retired colleague John Rice, that he had once come up with but never really published. In general, I was aiming for general tools that students could then apply to other cohomology theories that might come up in subsequent work. The focus on combinatorial examples and homological algebra was just so that there was something concrete to work with. The content is, I admit, a little non-standard, but I wanted something I would be happy with, not just one more rehash of Hatcher’s freely-available book (and others have mentioned to me something non-Hatcher would be good).

Because of the way AMSI approaches the content of the courses we all taught, I own the rights in the videos and the course content, and so I am releasing the videos of the course under a Creative Commons Attribution (CC-BY) license on YouTube:

I also typed up summary notes, and, because some students wanted it, some brief background notes on modules (both of these are CC-BY licensed as well)

Because of technical issues around the software and my low-spec laptop, the first few lectures do not have great video quality, but it **does improve** by about lecture 3. Oh, and in one lecture my generic tablet pencil, which I was loaned from my department, had its battery go flat, so I had to improvise somewhat. It only happened once, though!

On the whole I think everything worked reasonably well, aside from the intensity of the workload due to me underestimating quite how prepared I should have been. I ran a Discord server for student discussion (and memes), though the official assessment was ultimately all done in a Canvas instance, for security. The students (some auditing, some taking it for credit) came from a wide variety of backgrounds, from physics, maths, even computational neuroscience. So it was really interesting seeing people engage with the material in different ways, even grapple with concepts they had seen before, but done differently.

]]>“…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

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

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*!