I’m alive, just a bit antisocial – I’ve been spelunking in the wonderful caves of classifying spaces and group cohomology, which really got me to revisit my horrible algebraic topology fundamentals. I’m still a toddler in this field, but I think I’m beginning to take baby steps so I hope Haynes Miller and my other topology teachers (Anatoly, Dustin, Nick, etc.) would be eventually proud.
I just wanted to share a very short and simple insight today that made me very happy thanks to the Whitney sum formula: taking total Stiefel-Whitney classes are like exponentiating bundles (because taking total Stiefel-Whitney classes of a direct sum of two bundles becomes multiplication of the total Stiefel-Whitney classes of the individual bundles. This is “well duh” type of information to seasoned topologists, but to me this is exciting for several reasons:
- for storing information in my brain, which is among one of the least well-suited-for-math brains in the department. For some reason, this aphorism-ish idea suddenly made it feel like I could manipulate these guys a lot better. Thankfully too, as a few months ago they were completely mysterious to me and I need them to do some computations.
- for computation, this allows us to divide-and-conquer. This is the obvious “better bombs and banks” reason for mathematicians. For me personally, it is using the folk idea that we can associate representations of a finite group with a real vector bundle and then calculate the Stiefel-Whitney classes of the bundle, in which case the decomposition of representations into direct sums exactly corresponds with taking direct sums of our bundles!
- for crazy ideas, this means I can think of it as a kind of exponential generating function associated to my bundle, in the sense that we associate exponential generating functions to combinatorial structures in combinatorics. It may be interesting to think about what “inverse Stiefel-Whitney classes” may mean, or even proving combinatorial generating function formulae as “categorification” of playing with bundles! I haven’t quite seen any good examples of this, so I’d be happy to hear some, or maybe even make some.
- for the cool idea that math “forces itself to happen,” it is a good mental experiment to consider what classes can possibly have this exponential property, in the sense you can think of defining the exponential function via . Of course we have to play with the constants a bit – in our analogue this is just making sure the classes take the value on trivial bundles – but it ends up being quite a restrictive property. Segal and Stretch’s “Characteristic Classes for permutation representations” explores this kind of perspective.
Just a short breath to catch some air – it is a busy year where I have some more logistical duties and side projects. Back to the topological caves I go, though I really hope this kind of thinking would be helpful for at least one of you.