I removed this from the introduction:
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)
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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 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)
I'm not sure but I think it should be for the cohomology groups and not -- Cheesus 15:42, 2 April 2006 (UTC)
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 -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)
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)
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)
What about the homology of product spaces? Doesn't that deserve a mention? -- Raijinili ( talk) 04:05, 31 July 2008 (UTC)
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)
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)
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)
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? — Cheers, Steelpillow ( Talk) 19:40, 11 January 2015 (UTC)
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)
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)
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. — Cheers, Steelpillow ( Talk) 18:11, 18 January 2016 (UTC)
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?
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)
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". Such 'recursive' definitions, however informal, are unhelpful for exposition. — Preceding unsigned comment added by Commevsp ( talk • contribs) 20:23, 3 April 2021 (UTC)
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)
thanks, this is clearer now. happy new year to you! une musque de Biscaye ( talk) 13:22, 26 December 2021 (UTC)
In 'Informal examples' let , then .
In the next section 'Construction of homology groups', there is no reference to how is derived from X, nor how .
Does anyone have the missing pieces?
Darcourse ( talk) 18:49, 26 February 2022 (UTC)
In
/info/en/?search=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)
The drawing of cycles on the (hemispherical) projective plane 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 Chaikens 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? — Cheers, Steelpillow ( Talk) 16:25, 14 August 2023 (UTC)
I removed this from the introduction:
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)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
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 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)
I'm not sure but I think it should be for the cohomology groups and not -- Cheesus 15:42, 2 April 2006 (UTC)
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 -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)
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)
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)
What about the homology of product spaces? Doesn't that deserve a mention? -- Raijinili ( talk) 04:05, 31 July 2008 (UTC)
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)
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)
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)
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? — Cheers, Steelpillow ( Talk) 19:40, 11 January 2015 (UTC)
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)
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)
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. — Cheers, Steelpillow ( Talk) 18:11, 18 January 2016 (UTC)
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?
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)
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". Such 'recursive' definitions, however informal, are unhelpful for exposition. — Preceding unsigned comment added by Commevsp ( talk • contribs) 20:23, 3 April 2021 (UTC)
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)
thanks, this is clearer now. happy new year to you! une musque de Biscaye ( talk) 13:22, 26 December 2021 (UTC)
In 'Informal examples' let , then .
In the next section 'Construction of homology groups', there is no reference to how is derived from X, nor how .
Does anyone have the missing pieces?
Darcourse ( talk) 18:49, 26 February 2022 (UTC)
In
/info/en/?search=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)
The drawing of cycles on the (hemispherical) projective plane 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 Chaikens 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? — Cheers, Steelpillow ( Talk) 16:25, 14 August 2023 (UTC)