I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam.
There's one student who really likes topology and I'd like her to do something related to it as long as it can be done in time and it's interesting by itself (I don't want her to prove some isolated algebraic topology result that would only make sense in an algebraic topology course).
Any suggestions?
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I once had a compendium of Jan-Erik Roos in which derived functors was compared with derivatives: derivatives measured the deviation from being a straight line, while a derived functor (as I remember the Ext-functor) was measuring the deviation of a module from being a vector space. Of course this was very informally and I have never seen anything like that elsewhere. But Roos was studying homological algebra under Alexander Grothendieck in Paris, so the idea might come from him.
I don't know if this is interesting for you, but the big problem to introduce homological algebra ought to be the motivation.
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Here is a suggestion: identities among relations for groups and van Kampen diagrams. One source is this book. Also van Kampen diagrams occur in many books on group theory.
The topic allows lots of pictures and relates to group theory. Part of the motivation is "how to specify infinite objects, such as infinite groups?".
Also how do you find consequences of rules? An example is showing not only that $x^4=1$ is a consequence of the rules $x^2y^{-2}=1 $ and $y^{-1}xyx=1$, but also exactly how. You can find some amazing van Kampen diagrams on the web.
Such questions are also at the historic roots of homological algebra.
October 11: Have a look at the beginning of Chapter 3 of the book I link to (pdf available there) for the relation to the notion of resolution in the history of homological algebra. The word syzygy is relevant. I feel that the history of subjects is often neglected for a "need" to get on to the "modern" view; but then one wants to know how much the modern view has solved the original problems.
Also Section 10.3 on p. 341 presents a constructive method for resolutions. In the case of groups, this has been implemented by Graham Ellis at Galway (Homological Algebra Programming). For modules, you need methods for calculation, often done by Grobner bases.
I'll be interested to know if you and your student take this up or not!
She could probably define singular (co)homology and do a couple of applications in two hours, if she just stated the main technical theorems. Possibly axiomatic homology theory would be a beginning point suited to the abstract and algebraic pitch of the course. Something more interesting would involve spectra and triangulated categories, but that's surely too much for this context.