Talk:Homology (mathematics)

message: 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 Čech 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)

message: :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)
replies: text: 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)
thread_title:  Request for clarification 

message: I'm not sure but I think it should be H^n for the cohomology groups and not H_n -- Cheesus 15:42, 2 April 2006 (UTC)
thread_title:  Notation of cohomology groups 

message: 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 \Delta-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)
thread_title:  Use for Homology? Example Calculations? 

message: 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
Some words that might be helpful include: free abelian group and free groups, group quotients, topological space, continuous map, chain complex, chain maps.  For an example of a different construction in algebraic topology, look at fundamental groups.  These are simpler to construct but harder to compute. Particle25 (talk) 04:29, 17 July 2010 (UTC)
thread_title:  I don't get this 

message: 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)
replies: text: 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)
thread_title:  Section "Construction of homology groups" 

message: What about the homology of product spaces? Doesn't that deserve a mention? --Raijinili (talk) 04:05, 31 July 2008 (UTC)
thread_title:  Properties 

message: The construction incorrectly stated that C_0 is always zero.  I fixed this and added in the possibly nontrivial boundary map \partial_1 to the initial chain complex. Particle25 (talk) 04:30, 17 July 2010 (UTC)
thread_title:  editing construction section 

message: I'm trying to understand the distinctions between homotopy and homology, and unfortunately, this article isn't helping much.  I don't see a clear definition here of what homotopy is, rather, I see something that looks like an operational definition.  So far, I'm only getting a rather vague concept that the distinction has to do with the ability of the formalism to detect structure of holes, and that this is connected to an abelian or non-abelian nature, but how the formalism does that is completely opaque to me.  70.247.166.5 (talk) 15:26, 17 June 2012 (UTC)
thread_title: Homology and Homotopy

message: The italian page, it:Omologia (topologia) contains a fair amount of good material that should be added here. The maintenance tag requested this was removed, and since this article is in good shape, the notice seemed more appropriate on the talk page. 
I am particularly fond of its use of images, its detailed examples, and its applications section. JackSchmidt (talk) 21:05, 18 July 2012 (UTC)
replies: text: It's a good idea. I added an examples section inspired by the Italian page. --Mark viking (talk) 23:30, 18 March 2014 (UTC)
thread_title:  Italian page should be used to expand the english page 

message: Shouldn't the article mention that homology theories satisfying the Eilenberg–Steenrod axioms are canonically isomorphic? Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:44, 31 March 2015 (UTC)
thread_title:  Eilenberg–Steenrod axioms 

message: Shouldn't the sphere be a square, each arrow clockwise: right, double up (first two are rather arbitrary), double left (instead of right), down (instead of left). If you have the image clear in your mind, forget the middle two paragraphs.
Visually, you close up the square along the diagonal, and if you only glue the edges and not the inside, and blew it up from the inside like a balloon, it would, topologically, make a sphere. The first instinct I had was, this square given here (right, double up, double right, up) flips an image if you slide it through one of the edges as it pops back through the corresponding other one. That can't be right. Not on a sphere.
Easier yet, imagine a full circle, made of two half circles connected by their ends, with a disc filling the inside. Orient both half circles from one same end to the other, that means if you imagine one half is on top and the other is on the bottom, both half circles with the arrow pointed left to right, or both half circles pointed right to left, if you decompose each half circle each into two segments put end to end, a left segment and a right segment each, the top left segment corresponds to the bottom left segment and the top right to the bottom right.
The point is: The corresponding edges, the ones with identical arrows, must both be pointing away from the same common point, or both towards the same common point between the segments, the edges, that the arrows rest on. Right?
(Edit) or simpler correction, left edge arrow pointing up should be double, right edge arrow pointing up should be single arrow, that would make a sphere... wouldn't it?
:Thank you for spotting that (blush). The simpler correction also makes them all more consistent visually. I have now corrected it. If it does not appear immediately, you will need to clear your browser cache. — Cheers, Steelpillow (Talk) 07:27, 20 May 2016 (UTC)
thread_title:  Four ways of gluing the square S2 error? 

message: In order to make sense of the text, the reader needs to know what is meant in this setting by the term "hole". Yet (as also pointed out in a comment above), the current text does not offer any definition, not even an informal one. It may be hard to write about such an abstract concept, and it may not be realistic to expect the full content of the article to be understandable to someone whose familiarity with mathematics does not extend beyond the college level. But in this case I believe the current article also does not serve the reader who is familiar with topological spaces and manifolds, but not specifically with homology theory.  --Lambiam 08:13, 20 February 2018 (UTC)
thread_title:  What is a hole? 

message: Why is this word only 'in part' from Greek?  logy, 'word, reason' is from Greek as well.  The word is entirely from Greek. --142.163.195.111 (talk) 21:17, 1 March 2021 (UTC)
thread_title:   in part from Greek ὁμός homos "identical" 

message: "A boundary is a cycle which is also the boundary of a submanifold". Such 'recursive' definitions, however informal, are unhelpful for exposition.  — Preceding unsigned comment added by Commevsp (talk • contribs) 20:23, 3 April 2021 (UTC)
thread_title:  A boundary is a cycle which is also the boundary of a submanifold. 

message: In 'Informal examples' let X=S^1, then H_0(X)=\mathbb Z.
In the next section 'Construction of homology groups', there is no reference to how C(X) is derived from , nor how \ker(\partial_0)/\mathrm im(\partial_1)=\mathbb Z.
Does anyone have the missing pieces?
Darcourse (talk) 18:49, 26 February 2022 (UTC)
thread_title:  How do we get from X to C? 

message: In
https://en.wikipedia.org/wiki/Homology_(mathematics)#Surfaces
with the arrows on squares, there are 8 such variations. What happened to the missing 4?
Darcourse (talk) 15:42, 16 July 2022 (UTC)
replies: text: Presumably you are looking at the combinatorics of the various arrow positions? Once you take rotations and reflections, i.e. isomorphs, into account, there are just 4 distinct possible topologies. — Cheers, Steelpillow (Talk) 17:11, 16 July 2022 (UTC)
replies: text: [Update] I wonder if what I said is not quite right. Take the square for S2. Flip just one of the single arrows, say the right had one from up to down. This now glues up into P2, but the change cannot be made via a straight forward symmetry transformation of the S2 or P2 square. Flipping the bottom double-arrow as well would yield K2. That accounts for two of your missing variations, I think the rest will be down to symmetries, but I may still have missed something. Overall, these four manifolds are definitely exhaustive, but that "up to symmetry" in the article probably needs correction. I wonder if there is a decent source describing all this. — Cheers, Steelpillow (Talk) 08:38, 25 June 2023 (UTC)

text: Correction made. — Cheers, Steelpillow (Talk) 10:25, 25 June 2023 (UTC)
thread_title:  4 missing surfaces