Year Plans

by Guy Paterson-Jones

It’s been a long while since I’ve put any of the mathematics I’ve been doing online. Now that I’m in my third year of my undergraduate degree, and finding myself with more time to do what I love, I think it’s a fine time to begin anew!

Most of what I write will probably be notes or streams of thought.

I’ve spent a lot of time recently playing with algebraic topology and homological algebra. I’m working through Massey’s introductory homology textbook, which presents singular homology using cubes rather than simplices. This makes a lot of things easier – for instance, proving that one can subdivide cubes into smaller cubes is trivial. On the other hand, a rigorous display of barycentric subdivision for simplices is a tedious affair.

One thing that’s struck me is the use of some basic analysis in the proof of excision. A critical lemmas used is a shrinking lemma of sorts:

Lemma 1 (Shrinking Lemma) Let {{\mathcal U}} be a collection of subsets of a space {X} such that the interiors cover {X}. Consider the homology groups {H_n(X, {\mathcal U})} generated by all singular cubes contained within some {{\mathcal U}_i}. Then the inclusion {H_n(X, {\mathcal U}) \rightarrow H_n(X)} is an isomorphism.

This lemma essentially tells us that we only need to consider cubes that are ‘small’ in some sense to compute the homology. The proof works by pulling back the cover to a given cube, using compactness and then subdividing the cube enough times that each subcube lies within a set of the cover. The Lebesgue number of the cover tells us how many times is enough – this is the smallest {\epsilon \in {\mathbb R}^+} such that every ball of radius {\epsilon} is covered by some set in the cover. Of course, this relies on the fact that a cube is a metric space.

Given the combinatorial roots of homology, I would not be surprised if there was some way of bypassing this analytic argument. I would love to see a proof using something like the nerve of the cover here instead. A theorem I find quite pleasant in this sort of direction is the following:

Theorem 2 (Nerve Theorem) Let {{\mathcal U}} be an open cover of a paracompact space {X} such that every finite intersection of sets in {{\mathcal U}} is contractible. Then the geometric realisation of the Čech nerve of {{\mathcal U}} is homotopy equivalent to {X}.

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This is something I’d like to think more about in the future. Indeed, I love the interplay between combinatorics and topology that homology theory provides! A lot of combinatorial results like Sperner’s lemma have topological equivalents, such as (in this case) Brouwer’s fixed point theorem. The original proof of the Kneser conjecture, for instance, proceeded using homology. I’ll have to write about this stuff some time – it’s fascinating.

So what does this year look like for me, mathematically and otherwise?

First of all, I want to read some of the great texts in algebraic topology, and watch publically available lectures by the masters. This is due to a desire to seriously up the ante with my repertoire of tools this year. I plan to tackle most of the following:

  1. Topology from the Differentiable Viewpoint by Milnor.
  2. Morse Theory by Milnor.
  3. Characteristic Classes by Milnor and Stasheff.
  4. K-Theory by Atiyah.
  5. Differential Forms in Algebraic Topology by Bott and Tu.
  6. Algebraic Topology by Spanier.

These are all classic sets of lecture notes and/or books. As far as algebraic topology is concerned, they introduce a lot of the basic tools in various settings – I’m especially interested in the periodicity phenomena in K-theory, the crash course in Riemannian geometry nestled within Milnor’s notes, and learning a bit of cobordism. There’s a lot of overlap in this list, however, so I’ll probably be skimming here and there.

Another thing I’d like to get some familiarity with at some point (probably not this year) is surgery, {h}-cobordism and where we stand with the classification of {n}-manifolds. A book tying a lot of this together is The Wild World of 4-Manifolds by Scorpan, which now makes me associate manifolds with constellations.

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On the flip side, I’ve been perusing Weibel’s famous textbook, An Introduction to Homological Algebra. It’s certainly made the constructions in homology crystal clear to me, and I’m looking forward to when things get a whole lot more hardcore.

ADE classifications are all part of the game, and having finished up Erdmann’s Introduction to Lie Algebras, I think I should move onwards to Humphrey’s Introduction to Lie Algebras and Representation Theory. The story of simple Lie algebras is a fascinating one of analogy and connection. Also, my linear algebra has improved a crazy amount reading these.

Realistically, I doubt I’ll finish reading all of these books. But I hope I can at least make a stab at each of them. I’m thinking of LaTeXing up some solutions to exercises and posting them here along with notes. We’ll see how that turns out!

 

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