Talk:Algebraic topology
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Please Expand
Algebraic topology is such a great subject; I wish this article were more involved. I don't have the expertise to expand it, but I'm sure someone out there in wiki-land does. Could someone maybe beef this thing up a bit? User:Amp
- Category:Algebraic topology as a whole surely takes it a long way. Charles Matthews 08:52, 31 August 2006 (UTC)
New To Advanced Math
Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as algebraic topology, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006
I'm sure the claim that homology groups are always finitely generated is a bit overenthusiastic. What do we need to assume, maybe connected manifolds? AxelBoldt
The talk about torsion and orientability: when a space isn't a smooth manifold, orientability isn't any longer a naive concept.
Charles Matthews 10:45, 3 Dec 2003 (UTC)
In complement to Charles Matthews' comment:
1) The first homology group of a plane minus an infinite number of disjoint holes is definitely not finitely generated. I think it is safe to say that "for a complex of finitely generated chain groups, the homology groups are finitely generated."
2) The relation of torsion and orientability is indeed not very simple, and one should take some care: One can define the homology of topological spaces where the notion of orientability makes no sense. However, for e.g. simplicial complexes, one can define the notion of orienting cycle, but I don't know if there are simplicial complexes with such cycles which do not cover a topological space which is acceptable for a manifold. Moreover, the question is not simple even for manifolds: The Möbius band is an example of a non-orientable manifold with boundary with no torsion in its absolute homology groups, and the lens spaces (in Hatcher's book) seem to be orientable manifolds, even if their absolute homology groups have torsion. There is a theorem which states that "The absolute homology groups of compact, orientable manifolds with boundary which are embeddable into R^3 are torsion-free". If you ease any of the assumptions 1) "absolute" 2) "compact" 3) "orientable" 4) "embeddable into R^3", you can demonstrate a counterexample with torsion.
In reply to Saku Suuriniemi's comment:
1) The only condition for f.g. homology that I run into regularly is if the space is a finite-type CW-complex, and by this I mean that the complex has finitely many cells in each dimension. (Easy proof: cellular homology.) Certainly the claim you make in quotation marks is true (Z is noetherian).
2) The assertion about torsion and orientability is clearly false as you note. Example: RP3, 3-dimensional real projective space. However the lens spaces are not always orientable: the real projective spaces are all lens spaces, but RPn is orientable if and only if n is odd. So Hk(RPn) = Z/2, 0 < k < n, k and n odd is our counterexample. (Reference: Hatcher, AT, p. 144.) I feel I should point out, though, that we can always define orientability with respect to G-homology for any G, for any space. Moreover, since I see this topic has been dormant for a long time, I will go ahead and make the necessary changes.
Alodyne 05:20, 25 Mar 2005 (UTC)
Long time, no visit. I'm not sure what your point 2) concerns. The assertion about the existence of the theorem? If so, then are you sure RP3 embeds to R3? I doubt it. What comes to the orientability of the lens spaces, if you find even a single orientable lens space with torsion, it is a counterexample to the "orientable => no torsion" - "torsion => non-orientable" implication pair. The "non-orientable => torsion" - "no torsion => orientable" pair sank with the Möbius band. What I'm trying to explain is that the relation between the two is not at all simple.