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What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free. The deeper the methods used, the better. All the above can be proved with just the fundamental ...
There are also relationships between more classical algebraic geometry and algebraic topology, e.g. as in Hirzebruch's book and his proof of his version of the Riemann--Roch theorem. These sorts of connections don't seem to be as much in vogue right now (although I am not an expert in algebraic topology by any means, so may be wrong on this).
Algebraic topology, by it's very nature,is not an easy subject because it's really an uneven mixture of algebra and topology unlike any other subject you've seen before.However,how difficult it can be to me depends on how you present algebraic topology and the chosen level of abstraction.
$\begingroup$ For point-set topology, there are almost no pre-reqs, other than the ability to reason from definitions. Algebraic topology requires abstract algebra (it's not clear what you mean by Algebra, but it sounds like you are using the high school term, which is different from abstract algebra.) $\endgroup$ –
Questions tagged [algebraic-topology] Ask Question Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.
$\begingroup$ A researcher in algebraic topology could sometimes require A. Dold, lectures on algebraic geometry, because it uses very powerful techniques and covers very much in (co-)homology theory, but isn't easily read the first time — so less people like it. But if you learn how to read it, it should become a mighty tool.
Motivation: I am taking my first graduate course in algebraic topology next semester, and, up to this point, I have never taken the time to learn any category theory. I've read that category theory helps one to understand the underlying structure of the subject and that it was developed by those studying algebraic topology.
I hope this will help. I'm learning algebraic topology too. It's my first time checking content about simplicial complex. If there is anything wrong about the answer please comment. Thanks. Edit: I think I made a mistake. For the example $\sigma = \{1, 2\}$, the image of it has two choices. Maybe it's not well defined.
Trying to learn algebraic topology from May is like trying to teach yourself anatomy by reading a few medical journals. What book would work best for you to learn from depends on whether you prefer a geometric or an algebraic approach to the subject. For the former, there's Hatcher.
Directed algebraic topology is a branch of algebraic topology that has applications in concurrency theory when trying to avoid and resolve deadlocks and starvation. See for example here . Topological data analysis is an alternative to standard data mining, which allows one to infer global and structural properties about data.