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)[reply]

"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)[reply]

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)[reply]

While I am not an Algebraic Topologist, I am expert in nonsense, so that I can confidently answer: Yes, it is complete nonsense. Not only does it assume a complex of specialized technical notions, but it then proceeds to discuss them in ordinary language terms which simply do not apply -- they are the wrong ordinary language terms for the purpose. The inevitable result is incoherent babble. It is unclear whether C and D are single vertices or collections of vertices. The word "vertex" is not an ordinary language term. The relation of P and Q to C and D is left to be imagined. If C and D are collections, and happen to be disjoint, one might ask the alternative question: What could it mean to say that they do "join up" to each other. What does the word "before" mean in this case? It's clearly not an ordinary English use!

The addition of equally incoherent references will not serve to improve matters. The use of a link to "chain", which is an even more incoherent babble than is "Homology theory" in order to explain the terms used on this page leads to an exponential tangle of incoherence which requires an Ex Machina savior to unravel. Such is the prevailing fate of wikipedia mathematics pages: All too commonly, they serve more to obfuscate than to explicate. Aminorex 13:50, 21 June 2007 (UTC)[reply]

I have to agree. I came here to find a definition of Homology and found Gobledygook. Please don't identify something as "Simple" if it requires a Degree in Topology to comprehend. I can't give a simple explanation because this page doesn't make anything clearer to those of us looking to learn something from the Math pages of Wikipedia --Censorwolf 22:09, 19 October 2007 (UTC)[reply]

The very first paragraph of this article amounts to: "… homology theory is the … study of the … idea of homology. It can be broadly defined as the study of homology theories …" This doesn't make any sense, and instantly discredits the whole text. -- April 17, 2012 — Preceding unsigned comment added by 84.40.73.14 (talk) 16:16, 17 April 2012 (UTC)[reply]

I agree, can someone please clean this up? After all, it's a pretty important concept. Also, I don't understand why there are two article, this one and http://en.wikipedia.org/wiki/Homology_%28mathematics%29 . The latter article seems better btw. — Preceding unsigned comment added by 141.65.150.53 (talk) 08:53, 25 August 2012 (UTC)[reply]

Merge

This article has been merged into Homology (mathematics) and Cohomology due to a huge overlap; there was no material that was separate from these two articles. This article could be recreated in the future, but would need something new.Brirush (talk) 17:15, 18 October 2013 (UTC)[reply]