Is there something more to algebraic topology than detecting holes? 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)?
 A: 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.
