Talk:Homology (mathematics)
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I removed this from the introduction:
- Intuitively speaking, homology in the simplest case is the set of all possible non-equivalent non-contractible submanifolds (cycles) of a given manifold.
For one thing, homology isn't particularly about manifolds; one can do singular or Cech homology for any topological space, and some spaces aren't homology-equivalent to any manifold. (And of course there are more homology theories than those for topological spaces.) But more importantly, I don't see any way in which this statement is true, even when we restrict attention to manifolds. I can see how a cycle is a submanifold, but it doesn't have to be non-contractible; conversely, plenty of non-contractible submanifolds aren't given by cycles. And I don't see how homology can be a set (either of cycles or of certain submanifolds); at best, it's a sequence of sets (each set with a group structure that shouldn't be ignored). There may be something useful behind this sentence, but it needs to be made clearer. -- Toby Bartels 23:01, 12 Jun 2004 (UTC)
Request for clarification
- The procedure works as follows: Given the object X, one first defines a chain complex that encodes information about X. A chain complex is a sequence of abelian groups or modules A0, A1, A2... connected by homomorphisms dn : An -> An-1, such that the composition of any two consecutive maps is zero: dn o dn+1 = 0 for all n. This means that the image of the n+1-th map is contained in the kernel of the n-th, and we can define the n-th homology group of X to be the factor group (or factor module)
- Hn(X) = ker(dn) / im(dn+1).
As stated here, it appears that the chain complex can be chosen pretty arbitrarily with no dependence on X. Is this correct? Or, to rephrase my question, how does a chain complex encode information in X? Lupin 13:43, 8 Apr 2005 (UTC)