You may have noticed, if you fixate upon such things, that there has been an awful lot of “I’m going to…” and rather less “I have…” in this record. The poroblem, I feel, can be traced back to my youth, and the fact that I knew not of what I spoke. I arrived on a rather highish horse, you see, but have been punched in the throat by the fact that this stuff is pretty hard, and pretty lengthy.

Anyhow, I have now finished my first term of this game, and couldn’t honestly say that I haven’t learnt much, because the stuff is dribbling out of the old ears. The stuff I enjoy, it seems, is all rather old fashioned (as far as that can be, in a subject that, in it’s modern incarnation is about 60 years old) and so while interesting isn’t going to be very fruitful in terms of research material.

Doesn’t bother me though: I like what I like, and it’s early days, so I’ll stick to it for a while.

I’m off to Marbella for Christmas. See you there, if that’s your thing.

Excitingly, they were filming for “His Dark Materials: Northern Lights” on my street today, and I would have taken some photos, but the camera was batteryless. As far as I could overhear, it was just background shots, and so no shlabrities. I sat for a while and watched, to notice that it is most unglamorous and repetitive, this filmmaking lark. It also contains extended periods of standing around, and misunderstood requests:

“Dave, could you move that to the left, out of the way?”

“Here?”

“No, to the left, Dave.”

The magic of cinema.

Academically, I’ve been all over the place since last I wrote. However all these papers I’m supposed to be consuming don’t seem to dwindle. They are really quite technical, you see, and interesting, but require a fairish amount of working things out for yourself, which in turn requires all the Time there is.

So instead I flit hither and thither learning about things like principal G-bundles and characteristic classes, that you can actually show things with. I did some work on spectral sequences as well, but don’t have anything to apply them to yet, so I think I’ll wait until I do to learn more.

Unsettlingly, I don’t really know what I should be working on. This is a problem.

Lately I’ve been working on the papers below, but not perhaps as much as I should. I’ve been more interested in techniques to study the cohomology of Fiber bundles: the Leray-Hirsch theorem and the Gysin sequence.

It’s instructive to use Leray-Hirsch (or the Gysin sequence, for that matter) to compute the mod 2 and integral cohomology, respectively, of the real and complex Grassmannians, then define the Stiefel-Whitney and Chern classes to be the pullbacks of the obvious classes via a map that classifies the vector bundle. This is all well-defined, and some properties of the classes are immediate, but the Whitney sum formula for total classes is a bit tricky. I spent a goodish bit of Friday trying (and succeeding, thankfully) to prove it, with the vital guidance of George, a fellow topologist.

I don’t suppose it gives much more insight into the construction of these characteristic classes, but it avoids doing extra work.

There was also a talk on Friday by Eliana, another starting student of Ulrike Tillmann’s, on

C.F. Bödigheimer and U. Tillmann, Stripping and splitting decorated mapping class groups, Progress in Math. (78), Birkhauser (2001), 47-57 (link)

in which we covered approximately the first half. The second looks more complicated, and is mentioning configuration-spaces, so the other paper I am gradually traversing might come in useful. In fact, I may be giving a talk on it the Friday after this.

I’ve also been reading about Brown representability, as I had to give a quick talk on it this morning. Interesting as far as it goes.

I went up to Rhyl to visit my Grandparents this weekend, and found the above print on the wall of what is now an arcade, but used to be the “Left Bank Bistro” many years ago. Progress, what?

Well, pleasant reader, what have I been doing lately? Latest in the learning-fest is

G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. No. 34, (1968), 105-112 (link)

but just the classifying spaces bit for now. This paper isn’t where they were invented, that honour goes to Grothendieck, but it is perhaps where they were first explained in a simple manner. I had a brief talk about it with my supervisor today, seeing how natural things in a category give rise to natural things in its classifying space (which they should, seeing as the space is almost just a picture of the category), like a monoidal category has a monoid structure on it’s classifying space, and so on. Not entirely sure, so far, what the point of it all is, nor how it links in with the usual definition of the classifying space of a group (surely the classifying space of the category associated to a group must be the usual classifying space for the group?)

There is also a small project among the postgrad Algebraic Topologists in the department to read through and try to understand the paper

U. Tillmann, On the homotopy of the stable mapping class group, Inventiones Mathematicae Vol. 130, Issue 2 (1997), 257-275 (link)

(which may not be accessible to those on non-academic networks), but there are several others we think we’ll need to cover first.

I’ve also been doing the usual Hatcher-work, on homotopy theory this week, so loopspaces, homotopy construction of cohomology, fibrations, Postnikov towers and obstruction theory (which is something I’ve wanted to know about for ages, because it sounds so cool. However, now that I know about it I can’t think of anything nontrivial to apply it to. Ho hum).

Plan for this week:

• Mark my first set of scripts, for Professor Nigel Hitchin’s Geometry of Surfaces course, and attend the class bright-eyed, willing to help, and able to.
• Find a good exposition of (semi-)simplicial sets, read, digest, ruminate. I think I have found on, by May.
• With this knowledge, apply myself to the first paper mentioned above, while maintaining a beady eye on how it applies to the paper in my last post.
• Pick out choice morsels from the Extras Disc in Hatcher. Brown Representability, H-spaces, Limits and Ext should do.
• Read about things I’m already supposed to know about in this fantastic new book I’ve found, who’s name currently escapes me. It may be by May though, who for this week is my idol.
• Mark my second set of scripts, this time with the steely eye and grim resolve of an old hand, for Professor Marc Lackenby’s Topology and Groups course, and attend the class with optimism, as this should be something I’m quite happy with.

I had a brief talk with my supervisor yesterday to see what I’ll be up to. It looks like the short-term goal will be to understand

G. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 21 (1973), 213-221 (link)

but putting in the hours with Hatcher’s Algebraic Topology will carry on as usual. Last night I discovered Whitehead products on homotopy groups, and it was nice to know I haven’t forgotten everything. In fact, it looks like I’m meant to give a short talk on either Obstruction Theory or the Cohomology of Fibre Bundles. No doubt it will appear here if I do.