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At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher. A k-chain is thought of as a formal combination

   Σ aidi

where the ai are integers and the di are k-dimensional simplices on X. The boundary concept here is that taken over from the boundary of a simplex. This explanation is in the style of 1900, and proved somewhat naive, technically speaking.

Ummm...in English, for those of us with math ph.d.'s in something other than topology??

I agree... this needs to be rewritten, bearing in mind that the people reading it don't already know about the subject!!

Guys, that is the intuitive, geometric motivation. It is how homology was first conceived. The formal definition is less accessible. Charles Matthews 20:36, 11 September 2005 (UTC)

"simplex" then "semplices" and not "simplices". ahmedSammyH 13:16, 12 December 2005 (UTC)

I'm a topologist, and I think that this can be made much more intuitive: for example, the example of curves on a surface being homologous in terms of cobordism? This also lends itself well to pictures.

What I'm not sure is why we're trying to define "homology" as in "these two curves are homologous" on Homology theory. Wouldn't this explanation (hopefully clarified/simplified) fit better on Homology? I know it would require some restructuring there, but I think this concept needs a home, for all the readers out there who aren't algebraic topologists. Tesseran 03:42, 1 January 2006 (UTC)

[edit] Is this complete nonsense?

To single out one sentence (in bold) from the "simplest case" given in the "Simple explanation":

"At the intuitive level homology is taken to be an equivalence relation [on chains] [...]. The simplest case is in graph theory, with C and D vertices and homology with a meaning coming from the oriented edge E from P to Q having boundary Q — P. A collection of edges from D to C, each one joining up to the one before, is a homology. In general, a k-chain is thought of as [...]"

Again, what on earth does this mean? How exactly does a collection of edges from D to C correspond to an equivalence relation on chains? What does it mean for one edge to be "before" another edge, in a collection of edges? What would it mean for two edges from D to C not to "join up" to each other?

This wouldn't be so frustrating if there were references and external links. Buster79 23:36, 18 September 2006 (UTC)