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I have been reading in the last few months John Lee's Introduction to topological manifolds and Rotman's Introduction to algebraic topology. The reason I started learning AT is that it seemed like a beautiful and elegant theory, describing the properties of topological spaces.

However, as I come to think about it, it seems to me like I have been doing almost nothing but "detecting holes" with fundamental groups and homology groups and it's getting pretty boring. I also don't like the process of building spaces from simplicial simplexes or construction CW spaces.

My question is: Is there anything more to algebraic topology to offer? Are there more unique and beautiful theorems (like bursak-ulam or hairy ball)?

yoyo
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  • Topological Data Analysis builds out of Algebraic Topology and it lets you do all kinds of data analysis on large point sets – Alan Aug 16 '21 at 07:32
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    What algebraic topology books have you read? What theorems have you learnt from them? This may be a good way to self answer your question. – mathcounterexamples.net Aug 16 '21 at 07:33
  • As in every area of mathematics, of course there is an abundance of beautiful theorems ;) – Qi Zhu Aug 16 '21 at 07:38
  • What is a "beautiful theorem" ? I do not think that we can make this objective. – Peter Aug 16 '21 at 07:56
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    Algebraic topology studies properties of topological spaces via algebraic invariants. Those invariants are usually up to homotopy equivalence. "Holes" is not a precise concept, but typically you can "detect holes" because those "holes" prevent some deformations (meaning homotopy). From that point of view the answer is "yes", it is mostly about "holes". But the importance of those "holes" seem to be heavily underestimated by the OP. There are thousands of applications, often not obviously related to holes (e.g. Brouwer Fixed Point Theorem or Nielsen–Schreier theorem). – freakish Aug 16 '21 at 07:57
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    Why, yes. You can use cohomology to study intersections of subvarieties. Fiber bundles or vector bundles tend to correspond to some classes in (co)homology. There is also obstruction theory (you want to extend a continuous map from the low-dimensional skeleton of a CW-complex [sorry] to the full complex) where all relevant invariants are in some twisted homology groups. – Aphelli Aug 16 '21 at 08:58

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I wanted to briefly comment but wrote too much.

First of all, I want to defend holes a bit. I don't know about you, but I find it very intriguing that for instance $\pi_4(S^2) = \mathbb Z_2.$ Even if the end is to "detect holes", means can be very varied, profound and elegant. E.g. the above result can be shown in a few lines by the sneaky and somewhat enigmatic machinery of Serre killing, exhibited geometrically by using framed manifolds, or obtained by purely algebraic means of heavy Adams spectral sequence artillery. These are all doors to vast and mysterious worlds. In general, studying homotopy groups looks to me much more than merely "detecting holes". It has deep connections with both algebra and geometry. A bunch of random examples: there is Wu's formula (slide 5), a short, purely algebraic expression of $\pi_n(S^2);$ there is classification of exotic spheres, where recent computation of stable homotopy groups by Wang & Xu showed that odd-dimentional spheres with the unique smooth structure are only $S^1,S^3,S^5,S^{61};$ one of the classical "producers" of stable homotopy groups of spheres is the $J-$homomorphism, where you get beautiful geometry with vector bundles and $K-$theory.

Speaking of $K-$theory, there are invariants which are not "holes". There are cohomologies, which are rather obstructions than holes, or vector bundles. They also lead to "elementary" applications which you seem to like, for example: via $K-$theory it is easy to show that there are no division algebras over $\mathbb R$ in dimensions other than 1, 2, 4, and 8.

Building spaces from simplicial simplexes is indeed boring, but it becomes much more fun once you learn about purely combinatorial approaches to homotopy, where simplicial complexes enable one to introduce AT ideas in a lot of different contexts. It turns out that (commutative) diagrams that you draw can be themselves treated as spaces (see e.g. the nerve construction), from which a profound connection emerges between AT and many other branches of mathematics, all the way to number theory. You get higher categories, motivic homotopies/homologies, or you can even put homotopy intuition at the core of mathematical understanding and teach computers math this way (see Homotopy Type Theory).

There are also of course more classical connections, as with differential geometry, making AT so interesting to physicists.

I am sure experts could give you a never-ending stream of fancy theorems and sudden links to other domains. Of course "detection of holes" is one of the primary aims, but I see this rather as an advantage: with holes you ask a natural and understandable question instead of focusing on something which is hard to describe even to fellow mathematicians, or justify for yourself at least. I think this is exactly why AT ideas are now finding so many manifestations which no one could envisage.

Арсений Кряжев
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