Talk:Homology (mathematics)

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Mathematics rating:	Start Class	High Priority	Field: Topology
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 00:33, 23 June 2007 (UTC)

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)

Contents 1 Request for clarification 2 Notation of cohomology groups 3 Use for Homology? Example Calculations? 4 I don't get this 5 Section "Construction of homology groups"

[edit] 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 d_n : A_n -> A_n-1, such that the composition of any two consecutive maps is zero: d_n o d_n+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)

H_n(X) = ker(d_n) / im(d_n+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)

I added a reference that has a number of examples that help to clarify what a chain complex has to do with the space under consideration. Orthografer 22:38, 21 October 2006 (UTC)

[edit] Notation of cohomology groups

I'm not sure but I think it should be Hⁿ for the cohomology groups and not H_n -- Cheesus 15:42, 2 April 2006 (UTC)

[edit] Use for Homology? Example Calculations?

This page would benefit significantly from a discussion of the uses of homology, specifically in regards to the problem of classifying and distinguishing topological spaces. The introduction of the page suggests this project as a goal of the page, but, other than a single vague allusion to homology's usefulness for this purpose, no other comprehensive information is readily apparent on this page or any of the pages that it refers to.

As it stands, this page, and the pages linked to do a fine job of suggesting the breadth of ideas inspired by the study of homology (in the extensive classification of cohomology theories, the allusions to simplicial and singular homology, to homological algebra, etc.) without clearly showing, by means of detailed examples and references, what homology does for us or how it can be applied.

Hence I would offer the friendly suggestion that this page should at least link to (if not contain) some carefully worked out computations of chain complexes, their boundary homomorphisms, and the resulting homology groups for a variety of common spaces. Examples of how to construct spaces out of simplicial (or perhaps Δ-complexes, see Allen Hatcher, Algebraic Topology]) would also be very helpful. Mention of homology's behavior with respect to different notions of equivalence of spaces such as homeomorphism and homotopy-equivalence would also be valuable.

For a different perspective, compare this article to the articles on covering maps and homotopies to see how the abstraction in this article could be somewhat tamed. --Michael Stone 17:22, 9 April 2006 (UTC)

[edit] I don't get this

Please make a list of words that I would have to understand to get this and maybe put it at the top of the page. --149.4.108.33 00:37, 7 March 2007 (UTC)

[edit] Section "Construction of homology groups"

This section is pretty incomprehensible. I came to this article with a reasonable understanding of topology up to (but not including) homology, and this is the first thing I looked at to get an idea for it and I couldn't follow any of it.

Given an object such as a topological space X, one first defines a chain complex A = C(X) that encodes information about X.

The "chain complex" explained in the next couple of paragraphs doesn't appear to have anything to do with X. I can't make out any requirements on the chain complex that involve X in any way. Could someone who understands this stuff please give this section a rewrite? Maelin (Talk | Contribs) 13:35, 29 October 2007 (UTC)

How to obtain a chain complex from a topological space isn't described in this article - it's in the singular homology article. Rather, this article describes how the homology groups are obtained from a chain complex, regardless of how the chain complex arises (since they can arise from things other than topological spaces). This no doubt needs to be made clearer - I may attempt this later. --Zundark 14:07, 29 October 2007 (UTC)