Talk:Homology (mathematics)

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)

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)


 * 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)

I had exactly the same question as above. ToWit: In the section “Construction of homology groups” you presume some topological space X and then only mention it again in the context “Zn(X)” and “Bn(X)”. This notation needs definition or a reference to an article with a definition. This is the crux of the info that one seeks connecting the algebraic and topological. I think I need a definition, not an example! What does “Zn(X)” mean? —Preceding unsigned comment added by NormHardy (talk • contribs) 03:09, 28 October 2010 (UTC)

Notation of cohomology groups
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)

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 $$\Delta$$-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)

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

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)

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)

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

editing construction section
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)

Homology and Homotopy
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)

Italian page should be used to expand the english page
The italian page, 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)


 * It's a good idea. I added an examples section inspired by the Italian page. --Mark viking (talk) 23:30, 18 March 2014 (UTC)

Betti numbers and torsion coefficients
According to Richeson (Euler's Gem, The Polyhedron Formula and the Birth of Modern Topoplogy, Princeton University, 2008), Betti numbers and torsion coefficients are the defining topological invariants for manifolds. He introduces them as part of his very elementary discussion of homology, so I kind of expected this article to mention them too. There is an article on Betti numbers but nothing at all, anywhere, on torsion coefficients. I began to draft a stub of that missing article (here) but then I thought, why is it not mentioned anywhere? Where should it be covered? Would Torsion coefficient (geometry), Torsion coefficient (topology) or Torsion coefficient (homology) be the best title, or should it be included, with Betti numbers and other stuff, in an article about something else, say an Introduction to homology? &mdash; Cheers, Steelpillow (Talk) 19:40, 11 January 2015 (UTC)


 * Singular homology mentions the possibility of using coefficients in an arbitrary ring R, which would include the case of torsion coefficients Z/mZ. However, there's not much in that article besides the mention of the possibility.  Ozob (talk) 14:31, 12 January 2015 (UTC)


 * As far as I understand it (I'm not an expert), in simplicial homology the resulting homology groups are finitely generated. Finitely generated abelian groups can be written as the direct sum of a number of copies of Z, giving the Betti number, summed with the torsion subgroup, which is the direct sum of cyclic groups of the form Z/mZ. The idea of a torsion subgroup is a group theoretical one, but as you say, the coefficients have a homological interpretation. Probably the best existing article to discuss these is torsion subgroup with a section on topological applications, or perhaps simplicial homology with an expanded interpretation of the homology group structure. Torsion coefficients may be worth a mention in the simplicial homology section of this article, but going into detail here would probably be undue weight. Torsion coefficient (homology) might be the best new article title, as torsion coefficient already means something else in materials science. --Mark viking (talk) 22:25, 12 January 2015 (UTC)


 * It's a little more complicated than just the fact that simplicial (or singular, etc.) homology groups are finitely generated abelian groups. The use of coefficients changes the homology groups.  This is, at least in part, the point of the universal coefficient theorem: It asserts, for each $$i > 0$$, the existence of a short exact sequence:
 * $$0 \to H_i(X; \mathbf{Z}) \otimes \mathbf{Z}/m\mathbf{Z} \to H_i(X; \mathbf{Z}/m\mathbf{Z}) \to \operatorname{Tor}(H_{i-1}(X; \mathbf{Z}), \mathbf{Z}/m\mathbf{Z}) \to 0.$$
 * The tensor product in the first term converts the free part of the integral homology into copies of $$\mathbf{Z}/m\mathbf{Z}$$. Additionally, it will retain the $$\mathbf{Z}/m\mathbf{Z}$$ parts of any torsion factors that were already present.  The universal coefficient theorem says that this is not the same as the homology with torsion coefficients.  If there's $$\mathbf{Z}/m\mathbf{Z}$$-torsion in the integral homology in degree $$i-1$$, then it shows up in degree i in the homology with coefficients.
 * I don't think that torsion coefficients are overwhelmingly important. They have their uses (coefficients in a finite field are particularly helpful) but there really isn't much to say about them besides their existence and the universal coefficient theorem.  Ozob (talk) 02:22, 13 January 2015 (UTC)


 * Thanks for all the comments. There seems a huge gulf between the mindset of characterising real geometrical manifolds where it all began vs. the algebraic discipline based around Abelian groups. I think it best if I create a new stub article for my draft and don't attempt to merge it in anywhere myself. The article on Betti numbers introduces them in the context of algebraic topology, which is somewhat wider than homology alone, so I think it should probably be named Torsion coefficient (topology). &mdash; Cheers, Steelpillow (Talk) 16:57, 13 January 2015 (UTC)
 * Now done. &mdash; Cheers, Steelpillow (Talk) 14:28, 14 January 2015 (UTC)

Eilenberg–Steenrod axioms
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)

The usual problem
This problem needs a name, if it doesn't have one already; it's all too common on wikipedia mathematics articles. "Expertitis"? Expert blindness?

I came to the homology page to learn what it is, having been coming across the term today repeatedly. I didnt understand much at all of "homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group." So (can't expect the introductory paragraph to explain everything) I went further down the page.. Absolutely no luck.

I have the strong impression that the article is understandable only by - people who already know (a lot about ) what homology is. Or am I being told that I should keep away, this is for experts only?

I really can't object strongly enough to this problem in wikipedia writing. This page as it stands, though, I suppose, pleasing to the experts who painstakingly put it together, was totally useless in enlightening me in the least about the subject. And, I venture to guess, the vast majority who come to this page. I also guess that the majority of wikipedia mathematics articles have this gaping flaw.

Maybe wikipedia needs another layer of articles for people trying to learn about a subject. I would have thought that that should be the main purpose of wikipedia articles. Apparently the experts think not.

This type of article is written by experts (a good thing) and read without the non-expert reader in mind - that is the crux of the problem. I guess it would take a lot more time and effort than merely to write something that you can follow yourself. Or maybe its purpose is to keep outsiders out, not to welcome people in - to intimidate, not to communicate. 110.20.158.134 (talk) 04:21, 1 December 2015 (UTC)
 * Like many other mathematical topics, homology arose in the context of one discipline and turned out to be so useful in other disciplines that it became more and more abstracted. Homology was originally conceived by Henri Poincaré as a way of investigating and classifying topological manifolds according to their Betti numbers and torsion coefficients. In essence, it classifies the cycles - closed loops and stuff - that can be drawn on the manifold but not transformed smoothly into each other. Emmy Noether later abstracted and extended these ideas to create homology groups, the discipline of algebraic topology, and the foundation of the present Confusapedia article. (See Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, 2nd Ed. Princeton, 2008, pp.254-264.). I'll try to find time to do a little homework and add this to the article. &mdash; Cheers, Steelpillow (Talk) 10:46, 1 December 2015 (UTC)
 * P.S. I agree entirely about the arrogant "you have to be one of us before you can understand" claptrap. As every practising engineer knows, an "expert" is a drip under pressure. Worse, it is all too often used to obfuscate many an empty or even fraudulent technical paper written by such drips under pressure. The additional technique of adding dozens of equally empty or misleading citations tends to run hand-in-hand. At least this article has a way to go there, so there is hope for it yet. &mdash; Cheers, Steelpillow (Talk) 10:50, 1 December 2015 (UTC)
 * Now taken a first pass. Anything in particular still missing? &mdash; Cheers, Steelpillow (Talk) 12:02, 1 December 2015 (UTC)
 * I tried to rewrite the lead to be more intelligible. Let me know if it's an improvement.
 * Anon, I feel your pain. Reading mathematics on Wikipedia can be quite difficult. But I have long since decided that it's because writing mathematics is difficult. Those of us who understand something often don't do a good job of explaining because it's a lot of work, and sometimes it requires better writing skills than we may possess. I for one don't intend to write for experts, and I don't want to keep anyone out, but sometimes I fail.  The best solution is to let us know when something is no good.  That's the quickest way for us to learn it needs to be fixed.  Ozob (talk) 00:41, 2 December 2015 (UTC)
 * Thank you. It is always good to be reminded that there are folk more knowledgeable than oneself who are willing to listen and help. &mdash; Cheers, Steelpillow (Talk) 11:01, 2 December 2015 (UTC)
 * I appreciate all your efforts and intentions, I truly do, but I find the "informal examples" confusing on several points. (Note: I do have some formal knowledge of mathematics, although not of homology or most of its related subjects. So my impressions may be different from an even less informed reader's. ) Two items in particular: (1) I appreciate (from the lead) that "holes" are difficult to define formally, but the current text does not even attempt an informal definition. The passage in the talk page above that it's about "circles" that can or cannot be transformed into each other was much more enlightening (suddenly I understand how the sphere examples work). (2) It's a complete mystery how the associated groups are chosen. Why the natural numbers? Is this even explainable, or is it just "for technical reasons"? (Note: I am not one of the above anons.)
 * I also feel there may be a need for a Wikipedia-wide discussion of how subjects like deeply technical mathematics should be treated? At the moment, there is a strong tendency for every such subject to be given an "intuition" in terms of other aspects of the same technical subject. I fear this chain of intuitions may even be circular. 193.91.226.149 (talk) 14:53, 13 December 2015 (UTC)
 * I have been struggling too. I only found out since the above discussion that a "homology group" comprises all the cycles (the n-dimensional circles or loops) which can be smoothly transformed into each other. The number of such groups is an integer, hence the Betti numbers which count the groups are also integers. I think - I may have that not quite right.
 * There has been lots of discussion on the problem of overly-technical "introductory" material. As mentioned above, this generally reflects the difficulty of being both a good mathematician and a good writer. But in some cases it is more complicated than that. Homology has been a branch of algebraic topology ever since Noether formalised the aforementioned groups. For a hundred years the algebra has dominated the subject and its more intuitive roots in the drawing of shapes on geometric manifolds have been neglected. Recently, some good mathematicians who are also good writers have begun to correct this imbalance and publish books with pictures instead of equations in, and I draw on those I stumble across in my efforts here. &mdash; Cheers, Steelpillow (Talk) 15:50, 13 December 2015 (UTC)
 * The reason why the current text does not attempt an informal definition based on smooth deformations is that such definitions describe homotopy, not homology. Consider, for example, the complex plane punctured at 0 and 1.  We'll draw an interesting closed loop.  We start at the point 1/2.  Now move clockwise around 1, continue it to a clockwise path around 0, move down to cross the line segment (0, 1) and make a counterclockwise path around 1, continue it to a counterclockwise path around 0, and finally return to the point 1/2 to make a closed loop.  This loop cannot be deformed into the trivial loop in the twice punctured complex plane because it is homotopically non-trivial.  However it is homologically trivial because it loops around each puncture a net zero times (once clockwise and once counterclockwise).  More algebraically, if x and y represent clockwise paths around 1 and 0, then the loop is xyx&minus;1y&minus;1.  This is non-trivial in the fundamental group (the fundamental group is free on x and y), but because it's a commutator, it must be trivial in the first homology group (the first homology group is free abelian on x and y).
 * Geometrically, homology is defined in terms of cycles and boundaries. It's not a very intuitive definition, but your only alternative is to define it as the functor represented by an Eilenberg–Mac Lane space, which is even less geometrically appealing.  And the definition is quite subtle!  For instance, suppose that you are on a real analytic manifold and that you want to work with real analytic chains and boundaries.  All this means is that you want every set you consider to be locally defined by the vanishing of a power series.  That seems harmless, right?  But there is no known way to do this, and it is probably impossible in general.  If you relax your conditions slightly to allow semi-analytic sets, which are locally defined by vanishings and inequalities in power series, then the first proof that this is possible is due to Robert Hardt, Homology Theory for Real Analytic and Semianalytic Sets, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie, tome 2, no 1 (1975), pp. 107–148.  That's about a century after people first started counting holes and about half a century since there were rigorous definitions available in some cases.
 * The reason for the appearance of the integers is that the integers are the simplest non-trivial group. They are the free abelian group on a single element, so they map to any other group.  It is possible (and useful) to define homology with coefficients.  Homology with coefficients is related to, but not the same as, homology with integral coefficients.  A precise relation between the two is given by the universal coefficient theorem.  A very common technique in modern algebraic topology is to work "locally at a prime p".  In practice, this means working with coefficient groups like the finite field Fp and the p-adic integers Zp.  Perhaps the phenomena that interest you are intrinsically mod p, or perhaps it suffices to look mod p for all p, or maybe the integral homology is just too hard. The universal coefficient theorem tells you how to relate calculations of integral homology mod p to p-torsion and to mod p homology, helping you to answer your question.  (Even more exotic coefficient rings are possible, but then the universal coefficient theorem becomes a spectral sequence instead of a short exact sequence, so extracting what you want becomes much harder.)  Ozob (talk) 17:44, 13 December 2015 (UTC)
 * I did say that I might not have got it quite right. Cycles originated in the idea of cuts: if you cut a manifold across, will it fall into two pieces? Betti numbers describe how many cuts you can make without it falling apart. For example you cannot cut a sphere at all but you can cut a torus twice. Cycles are in essence just the cuts marked out but not actually made, although they can do more complicated things like wind round twice, add to each other or cancel each other out. I am afraid that much of Ozob's post merely emphasises the gap between the algebraic sophistication and the simple-minded visitor. For example the idea of non-integral coefficients seems lost in the algebra (are they at least rational?). Anyway, I evidently still don't understand homology groups. For example if we have two cycles a and b on a torus, how can we tell whether they belong in the same group? Is that even a sensible question? Similarly, if a and b intersect, then is say a + b a member of a different group or does that depend on the value of a + b (which might be zero), or what? &mdash; Cheers, Steelpillow (Talk) 19:02, 13 December 2015 (UTC)
 * Any coefficient group or ring we please can be used. So we can have Z coefficients, Q coefficients (the rational case), Z/2 coefficients, and so on.  It is even possible to vary the coefficients over the space (see local system – this turns up when studying differential equations).
 * Geometrically, you can understand a + b to mean "cut along both a and b". When you have coefficients, though, you may want to cut with multiplicity (for instance, if a = b), and the geometry of that operation is not obvious.  Negative coefficients mean "glue" (the geometry of this operation is again not obvious, but if a and b have opposite multiplicities, then the meaning of cutting followed by gluing is to do nothing).
 * In a modern treatment, there is no such thing as a cycle until there is a group to put it in. First we define a group of chains; these are usually formal linear combinations of good geometric objects.  In simplicial homology, the good subspaces are subcomplexes; in singular homology, they are images of simplicial complexes.  We make our definitions so that chains come with a meaningful geometric notion of boundary operator.  The group of chains inherits a boundary operator, and we require that the boundary operator be a homomorphism satisfying the usual $$\partial^2 = 0$$ condition.  Once this is done, a cycle is a chain whose boundary is zero; that is, the group of cycles is the kernel of the boundary operator, and so it is clearly a subgroup.
 * This could be considered unsatisfying, because a cycle really ought to mean something that cuts up the space, and instead I talked about formal linear combinations and boundary operators. But I don't know how else to make things rigorous.  There are more or less geometric ways of discussing homology, but some amount of algebra is necessary because homology is an algebraic crutch to understand geometry.  You eventually have to admit that homology is an algebraic mystery.  It is a way of measuring a space, but its meaning is not clear.  It is as Michael Atiyah once said: "Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."  Ozob (talk) 21:45, 13 December 2015 (UTC)
 * All I need to do now is understand what is meant by "coefficient", "coefficient group", "subcomplex" (of what?), "image" (of a simplicial complex), "boundary operator" and "kernel" (of the boundary operator). Poincaré took to chains because the more intuitive treatment of cycles was not rigorous, and Noether turned it all into algebra. Once in a while some luminary comes along, who can turn the algebra back into intelligible geometric ideas. I have some introductory books and one on Calabi-Yau manifolds but there is a big gap in between where I flounder hopelessly. I think that two separate articles might be useful, an introductory one giving the intuitive geometrical version and a more comprehensive one giving the rigorous algebraic version. Does that seem like a good idea? &mdash; Cheers, Steelpillow (Talk) 12:33, 12 January 2016 (UTC)
 * Well: The group of chains is usually the free abelian group on some set of geometric objects. We get chains with coefficients in a ring R by taking the free R-module on that set.  And we get cycles, boundaries, and homology with coefficients in R by applying the usual definitions.  There's no content here, only definitions: To have coefficients in R means you used a free R-module to start with instead of a free abelian group, nothing else.  R can be any commutative ring (maybe even some noncommutative rings? I don't know).
 * By "subcomplex" above, I meant a subcomplex of a simplicial complex. That is, D is a subcomplex of a simplicial complex C if D is a simplicial complex which is a union of simplices of C.  Perhaps the confusion is that "complex" and "subcomplex" have other, algebraic meanings.
 * By "image of a simplicial complex" I meant its image under a continuous function. If C is a simplicial complex and X a topological space, and if f : C &rarr; X is continuous, I just meant f(C).  Though in retrospect, I really meant f, because the way in which the embedding is done matters (you want to know where the faces of the simplex are).
 * The boundary operator is defined differently depending on which homology theory you're discussing. For theories defined by simplices, it's the alternating sum of the faces.  Similar formulas turn up in many other homology theories.  (There are good reasons why.  See monad (category theory).)  The boundary operator is a morphism of complexes (complexes in the algebraic sense), and it has a kernel in the usual algebraic sense (things which get sent to zero).
 * As for what to do with the article, I'm not sure. There's some good, intuitive geometry here, but ultimately math has to be rigorous, and nobody has ever found a wholly geometric interpretation of homology (when expressed in terms of cycles and boundaries; there's a geometric interpretation in terms of Eilenberg–Mac Lane spaces).  So while the article should start with geometry and pictures, at some point it's going to be taken over by algebra.  It has to be, unless you can revolutionize our understanding of homology.  And I think it's a disservice to the reader to pretend otherwise.  Ozob (talk) 13:44, 12 January 2016 (UTC)


 * If you swap "usual" for "incomprehensible" you will get an idea of how much of that I can understand. Thank you for trying, anyway. I agree entirely that the intuitive geometry needs a strong caveat as to its lack of rigour. rather, I was wondering whether it might be better split off as a separate article called "introduction to homology" or "origins of homology" or similar. I have some diagrams I could add, showing cycles on various surfaces, such as the one here on the right. Or, I could add it all to the current section on "Informal examples". &mdash; Cheers, Steelpillow (Talk) 14:40, 12 January 2016 (UTC)
 * That's a good picture. Yes, pictures like that certainly belong in the article.  And there's precedent for "introduction to" articles (general relativity and introduction to general relativity).  For the moment I think it's best to expand this one, since it's not overly long (yet), but that's certainly an option for later.  Ozob (talk) 13:25, 13 January 2016 (UTC)
 * Well, I went for it. Not sure if "History" is the best place for it, but it's where the narrative slotted in most easily to the article as written. &mdash; Cheers, Steelpillow (Talk) 18:37, 13 January 2016 (UTC)
 * I like it. I made some edits: I defined homology classes. I tried to distinguish between 0-cycles, 1-cycles, and 2-cycles. "Smoothly" is the wrong word because it implies differentiability, but differentiability is irrelevant here. It's important that a homology class can be broken apart into sums of other classes. For instance, the homology of the twice punctured plane is Z^2, but its fundamental group is the free group on two elements, and this really comes down to the fact that in homology, I'm allowed to break loops in the middle, whereas in the fundamental group I can't.
 * What do you think? I hope I didn't put in too much jargon. Ozob (talk) 15:18, 18 January 2016 (UTC)

First, thank you for sanity-checking it and fixing/extending some things. While it is true that early workers in topology used the idea of gluing and cutting, that was before homology came along to unify the field. Both Richeson and Yau introduce homology itself in terms of "loops", called "cycles", which are "drawn on" a surface and can then be manipulated. So I followed their example in my version of the introduction and I think we should restore that. Rather than say, "A sub-manifold (oh, and on a surface that's just a loop")", we should say "a loop (oh, and in higher dimensions that generalizes to a sub-manifold)". Also, the idea of cutting and re-gluing manifolds to create new surfaces is not central to the underlying ideas but arises from them. At present it is stitched in and out of the narrative explaining the illustrations and I think that is unhelpful - it needs its own narrative and illustrations. As such it should be introduced afterwards - if at all. because I don't personally know whether it even counts as "homology", or whether the homology only tells you what you ended up with. &mdash; Cheers, Steelpillow (Talk) 18:11, 18 January 2016 (UTC)
 * I'm only familiar with the modern viewpoint on these things. What Euler, Riemann, and Betti thought they were doing is just something I've never studied (though I do have the impression that they were primarily concerned with surfaces). Nor am I sure about the right way to present these ideas. I agree that cutting and gluing isn't central, but I'm at a loss to explain homology in other elementary terms. You have to allow some amount of cutting and gluing in order to distinguish the first homology group from the fundamental group.
 * I think it might be better (but a lot more work) to use a whole different approach. Instead of using the first section of the article to describe homology, we should use it to describe history. We ought to say that Euler introduced the Euler characteristic and talk a little about the relationship between Euler characteristic and shape (not in as much detail as the Euler characteristic article, of course). Then describe what Riemann did, then describe what Betti did, and so on. Within a few paragraphs we reach Poincare, at which point we can be said to be talking about homology proper. If done properly, the reader will have some vague idea (not by any means a precise one) what a cycle and homology is. Then the current first section of the article could (with only minor changes) become the second section of the article. It ought to be easier for the reader this way.
 * Unfortunately, I'm not qualified at all to talk about history, so I can't help with such a reorganization. But I think it would be the right direction. Ozob (talk) 21:38, 18 January 2016 (UTC)
 * The approach you propose is pretty much that taken by Richeson. The problem here is that he takes seventeen chapters - about three-quarters of a decent-sized book - to go from Euler to Poincarè at a similar introductory level. Only his last chapter is about homology. So I think we have to refer that more detailed treatment out to another more general article, such as Topology or an Introduction to topology. Another way of looking at the treatment of cutting and gluing surfaces to change their topology is that it has its own algebra, which is a different algebra from that of manipulating homology cycles on an uncut manifold. Orientable and non-orientable homologies have variant algebras already and although I have hinted at this it needs expanding on. To mix in a third, entirely different algebra at this level is not helpful: it needs to be addressed in a later section (assuming it is still homology at all). Also, besides correcting some of my slips you have introduced others of your own, for example the two-holed torus is constructed from an octagon not a hexagon, and the table includes the 1-manifold or circle as well as surfaces. So I hope you will not be too offended if I unpick some of your changes to my version and try to merge the best of both of us in a different way. &mdash; Cheers, Steelpillow (Talk) 07:08, 19 January 2016 (UTC)
 * Now done. I have managed not to cut any of your significant additions, while also trying to introduce terms like "class" in an intelligible way. &mdash; Cheers, Steelpillow (Talk) 08:34, 19 January 2016 (UTC)
 * Thanks for finding and fixing my errors. I caught a few more: Cycles are not just low dimensional (the fundamental class of an n-manifold, for instance, is an n-cycle); the zero cycle is a particular cycle and there's only one of it (in simplicial or singular homology, it's represented by the empty set), but there are many cycles homologous to zero; cutting along a cycle homologous to zero can separate the manifold into more than two components (consider cutting along a figure eight on the sphere); Betti numbers and torsion coefficients are properties of manifolds, not cycles; the Euler characteristic is the alternating sum of the Betti numbers, so Betti numbers are more properly described as a refinement of the Euler characteristic than a generalization to higher dimensions. Ozob (talk) 13:49, 19 January 2016 (UTC)
 * One more comment. You say that it takes Richeson seventeen chapters to cover the history. But the present article spends only a handful of paragraphs. There is plenty of room for a middle ground. At algebraic K-theory I managed to write an extensive history (with the help of Weibel's article), long enough that some of it might deserve to be separated into its own article. But it's still not as long as a book or even Weibel's article, and I think it's a pretty good introduction to the subject. Something similar ought to be possible here. Ozob (talk) 14:05, 19 January 2016 (UTC)
 * I'd say, go to it and see how things shape up. I am signing off this work now, as I am starting a long wikibreak to go find out if the real world still exists. Thank you once more for all your help. &mdash; Cheers, Steelpillow (Talk) 17:46, 19 January 2016 (UTC)
 * As a third party observer, I think the work that resulted from this discussion has noticeably improved this article, imho. Woscafrench (talk) 15:36, 2 September 2017 (UTC)

Four ways of gluing the square S2 error?
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. &mdash; Cheers, Steelpillow (Talk) 07:27, 20 May 2016 (UTC)

What is a hole?
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)

in part from Greek ὁμός homos "identical"
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)

A boundary is a cycle which is also the boundary of a submanifold.
"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)

orientability of disk and ball
Table "Topological characteristics of closed 1- and 2-manifolds" says that E2 and E3 are not orientable. Is that correct? Also, looking at the source of the table that seems not the case @Steelpillow thanks! une musque de Biscaye (talk) 09:43, 25 December 2021 (UTC)
 * The Season's Greetings to you. Those manifolds were added to the table by in this edit a year ago. Unlike the others they are bounded manifolds, and the application (or otherwise) of homology to such manifolds is not explained in the article. I think it best to remove them. Also, I think the term "closed" is causing its usual confusion, so I will retitle the table accordingly. &mdash; Cheers, Steelpillow (Talk) 10:24, 25 December 2021 (UTC)
 * Now titled as closed and unbounded. For what it's worth:
 * A finite disc including its boundary is (geometrically) closed but also bounded.
 * A finite disc excluding its boundary is unbounded but (geometrically) open (as also is the Euclidean plane).
 * &mdash; Cheers, Steelpillow (Talk) 10:37, 25 December 2021 (UTC)

thanks, this is clearer now. happy new year to you! une musque de Biscaye (talk) 13:22, 26 December 2021 (UTC)

How do we get from X to C?
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 $X$, 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)

4 missing surfaces
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)


 * 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. &mdash; Cheers, Steelpillow (Talk) 17:11, 16 July 2022 (UTC)


 * [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. &mdash; Cheers, Steelpillow (Talk) 08:38, 25 June 2023 (UTC)
 * Correction made. &mdash; Cheers, Steelpillow (Talk) 10:25, 25 June 2023 (UTC)]

Drawing of cycles on the projective plane
The drawing of cycles on the (hemispherical) projective plane $$P^2$$ has recently been changed.

This new version breaks the consistent convention in all the accompanying drawings, that say B and B' are images of the same point. Instead it treats them as two coincident points on cycle b. This is also now at variance with the text, which relies on the previous consistency to carefully explain the situation as it was. I also find it overly cluttered and even less understandable than before. In this edit I therefore restored the original and asked for discussion here. However has chosen here to restore the inconsistent version once more, without discussion on this talk page. So I think we need an independent voice or two now. Which version should we include here? &mdash; Cheers, Steelpillow (Talk) 16:25, 14 August 2023 (UTC)


 * In none of the other pictures are there individual point images that are identified, so I didn't add any inconsistency--the convention that A and A', etc. are identified was used nowhere else. Of course, the arrow notation was used in other figures to identify line segments in particular orientations. Hours after I edited the picture, I edited the referring text https://en.wikipedia.org/w/index.php?title=Homology_(mathematics)&oldid=1170405942 to be consistent with the picture and to explain what is going on more. A and A' in the original are really the same point. It is not clear in the original also which of the two points under the B' are referred to by B'.  It is critical that the two places where the parallel branches of curve b meet the rim are recognized as two different points.  Other voices with ideas to better present this stuff to beginners be welcome to me too! Cheers, Chaikens. Chaikens (talk) 03:00, 15 August 2023 (UTC)
 * On thinking about this a little, I realize a problem with this article is that it may discuss historical conceptual origins of homology theory that later got elucidated into homotopy theory when algebraic topology was developed better. The idea that a cycle can be shrunk is of today's idea of homotopy. The section we are discussing gives homotopic arguments for why the four surfaces are different.
 * I think the homotopy (that is, continuous deformation of the double loop b that makes it into a single point) that shows "b+b=0" is more complicated than what the text describes. One of the two branches would be moved down so that it coincides with half of the rim.  The arrow on that moved branch would continue go back-to-forward from the viewer's perspective.  Then trying to move the branch below the rim would actually make it appear coincident with the other half of the rim--but the direction of the arrows would be forward-to-back.  It can the be brought up the other side of the hemisphere---and it will be directed opposite the other branch.  One point meeting the rim can then be brought up above the rim, and we get a loop like a, which is clearly deformable into a point.   Chaikens (talk) 04:20, 15 August 2023 (UTC)
 * Silly me, of course these points are in this diagram only. But you did not update the text until some hours after I had posted here (and I think it is still not quite adequate). The key feature of b is that it is double-wound on the same line, just as a and -a are on the Klein bottle. Consequently, your points B and B' are coincident. I did try all these added labels originally, but judged the diagram overly cluttered and the finer details not really relevant to the level of discussion. This is where we need other voices to form a consensus.
 * Your point about homotopy intrigues me. I originally added these drawings to the article on topology, which is where they appear in many introductory sources. Other editors then agreed to move it here. You are saying in effect that this is still the wrong place and that homotopy is the best home. Personally I think that we should be being less precious about it and doing what reliable sources do at this introductory level, which is to present them, along with the accompanying historical origins, under the heading of topology. So revisiting the consensus on that, preferably putting WP:RS over our editorial opinions, would be useful too. &mdash; Cheers, Steelpillow (Talk) 08:44, 15 August 2023 (UTC)