Talk:Algebraic topology

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JDBowen, this looks like a great start on reworking this article. Thanks! In case you haven't seen it, we also just rewrote all of fundamental group, which is still a work in progress. Feel free to jump in there. We can probably push the definition of pi_1 to that page only. I'm thinking about writing a new article for homology that's more accessible -- do you have any thoughts about what to include there?

I'm going to modify the statement that there are "two approaches to the study of homology" -- there are many others (even for topological spaces), like singular and (for CW complexes) cellular homology. They turn out to give you the same groups as a consequence of the fact that they all satisfy the Eilenberg-Steenrod axioms. And de Rham is really a cohomology theory, perhaps a minor distinction here. There are other cohomology theories too, like Cech cohomology and the cohomology of various sheaves that also give the same thing. Here's an article about some more. But this doesn't belong on the main page here, so I'll just delete "two". (: --MarkGall 00:14, 6 July 2009 (EDT)

Thanks! Let me just say I'll totally defer to you on any math changes you want to make, since you're a graduate student already and I am only starting my masters in the fall. Having said that, in regards to your question about describing homology - that's a toughie. My understanding is that Andy wants mathematics material to be accessible to non-major undergrads. I think homotopy is simple enough for anybody who has taken calculus, or even pre-calc, but homology is totally different. A chain of articles (no pun intended) would be needed to describe just singular homology, and as you pointed out, that is just one example, albeit an important one. I would be interested to see any ideas you might have for this article!JDBowen 07:30, 7 July 2009 (EDT)

My feeling is that the article should cover nothing except simplicial homology, except maybe to mention that there are other viewpoints. It probably ought to give a definition and then cover in great detail the examples of a sphere and torus (and some others we ought to think about) with computations on explicit triangulations. We should also work on a fluff paragraph or two about "n-dimensional holes" or some such to motivate it, and that's perhaps the most important part. And please don't defer to me on math changes -- I'm just a year ahead of you, and I'm sure I'll make many mistakes and confusing statements! --MarkGall