1558:, 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
1600:, 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,
2112:
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. â Cheers,
2199:-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.
2132:
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.
1505:
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.)
1820:
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."
29:
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.
1678:
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
118:
108:
87:
2158:. 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
1835:
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? â Cheers,
54:
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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.
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as well would yield K. 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. â Cheers,
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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. Â --
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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
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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
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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
2058:
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
1940:
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
1834:
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). PoincarĂŠ took to chains because the more intuitive treatment of cycles was not rigorous, and
1782:
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
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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
28:
For one thing, homology isn't particularly about manifolds; one can do singular or Äech 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
2149:
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
1677:
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.
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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
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Anon, I feel your pain. Reading mathematics on
Knowledge 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
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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
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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
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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
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I wonder if what I said is not quite right. Take the square for S. Flip just one of the single arrows, say the right had one from up to down. This now glues up into P, but the change cannot be made via a straight forward symmetry transformation of the S or P square. Flipping the bottom double-arrow
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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
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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
2007:
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
1508:
I also feel there may be a need for a
Knowledge-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
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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
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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 á˝ÎźĎĎ homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical
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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
1607:
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,
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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
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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
2217:
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
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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
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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
502:
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
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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
1504:
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
2404:
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. â
2111:
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
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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
1819:
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
1941:
of
EilenbergâMac 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.
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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
1534:
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
2059:
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.
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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
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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.
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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.
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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
592:
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?
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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.
2008:
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". â Cheers,
1604:, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie, tome 2, no 1 (1975), pp. 107â148. 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.
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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.
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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
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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.)
1281:
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
887:
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
2275:
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.
583:
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.
2128:
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.
743:
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
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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?
394:(X)â. 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 âZ
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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.
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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.
2295:
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. â Cheers,
1392:
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
2823:. 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
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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.
2232:
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. â Cheers,
2702:
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. â Cheers,
2380:
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!
1372:
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.
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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. â Cheers,
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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?
2213:
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
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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
1105:{\displaystyle 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.}
531:]) 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.
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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.
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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
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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? â Cheers,
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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
852:. 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
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I had exactly the same question as above. ToWit: In the section âConstruction of homology groupsâ you presume some topological space X and then only mention it again in the context âZ
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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.
2909:, 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
2291:(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?
1937:.) 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).
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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.
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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.
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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. â Cheers,
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58:
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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.
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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. â Cheers,
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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.
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1771:), and the geometry of that operation is not obvious. Negative coefficients mean "glue" (the geometry of this operation is again not obvious, but if
2350:"A boundary is a cycle which is also the boundary of a submanifold". Such 'recursive' definitions, however informal, are unhelpful for exposition.
1650:, or maybe the integral homology is just too hard. The universal coefficient theorem tells you how to relate calculations of integral homology mod
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1357:
This problem needs a name, if it doesn't have one already; it's all too common on wikipedia mathematics articles. "Expertitis"? Expert blindness?
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Intuitively speaking, homology in the simplest case is the set of all possible non-equivalent non-contractible submanifolds (cycles) of a given
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introduces them in the context of algebraic topology, which is somewhat wider than homology alone, so I think it should probably be named
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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.
1849:
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
2022:
That's a good picture. Yes, pictures like that certainly belong in the article. And there's precedent for "introduction to" articles (
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usefulness for this purpose, no other comprehensive information is readily apparent on this page or any of the pages that it refers to.
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The reason why the current text does not attempt an informal definition based on smooth deformations is that such definitions describe
2030:). 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.
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imbalance and publish books with pictures instead of equations in, and I draw on those I stumble across in my efforts here. â Cheers,
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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.
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Thanks for finding and fixing my errors. I caught a few more: Cycles are not just low dimensional (the fundamental class of an
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1585:), but because it's a commutator, it must be trivial in the first homology group (the first homology group is free abelian on
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I have been struggling too. I only found out since the above discussion that a "homology group" comprises all the cycles (the
1425:, 2nd Ed. Princeton, 2008, pp.254-264.). I'll try to find time to do a little homework and add this to the article. â Cheers,
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Why is this word only 'in part' from Greek? logy, 'word, reason' is from Greek as well. The word is entirely from Greek. --
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be the best title, or should it be included, with Betti numbers and other stuff, in an article about something else, say an
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As a third party observer, I think the work that resulted from this discussion has noticeably improved this article, imho.
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Given an object such as a topological space X, one first defines a chain complex A = C(X) that encodes information about X.
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on a torus, how can we tell whether they belong in the same group? Is that even a sensible question? Similarly, if
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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. --
2103:
First, thank you for sanity-checking it and fixing/extending some things. While it is true that early workers in
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1929:, because the way in which the embedding is done matters (you want to know where the faces of the simplex are).
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but not the same as, homology with integral coefficients. A precise relation between the two is given by the
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parts of any torsion factors that were already present. The universal coefficient theorem says that this is
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A finite disc excluding its boundary is unbounded but (geometrically) open (as also is the Euclidean plane).
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homotopy. The section we are discussing gives homotopic arguments for why the four surfaces are different.
1413:- closed loops and stuff - that can be drawn on the manifold but not transformed smoothly into each other.
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As stated here, it appears that the chain complex can be chosen pretty arbitrarily with no dependence on
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but nothing at all, anywhere, on torsion coefficients. I began to draft a stub of that missing article (
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by applying the usual definitions. There's no content here, only definitions: To have coefficients in
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to this page. I also guess that the majority of wikipedia mathematics articles have this gaping flaw.
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How to obtain a chain complex from a topological space isn't described in this article - it's in the
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Thank you. It is always good to be reminded that there are folk more knowledgeable than oneself who
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The tensor product in the first term converts the free part of the integral homology into copies of
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I am particularly fond of its use of images, its detailed examples, and its applications section.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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1761:". When you have coefficients, though, you may want to cut with multiplicity (for instance, if
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have opposite multiplicities, then the meaning of cutting followed by gluing is to do nothing).
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means of detailed examples and references, what homology does for us or how it can be applied.
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1736:/2 coefficients, and so on. It is even possible to vary the coefficients over the space (see
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Your point about homotopy intrigues me. I originally added these drawings to the article on
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changes to my version and try to merge the best of both of us in a different way. â Cheers,
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1892:. Perhaps the confusion is that "complex" and "subcomplex" have other, algebraic meanings.
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with the arrows on squares, there are 8 such variations. What happened to the missing 4?
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I tried to rewrite the lead to be more intelligible. Let me know if it's an improvement.
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1857:-module on that set. And we get cycles, boundaries, and homology with coefficients in
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later abstracted and extended these ideas to create homology groups, the discipline of
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By "image of a simplicial complex" I meant its image under a continuous function. If
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In the next section 'Construction of homology groups', there is no reference to how
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https://en.wikipedia.org/search/?title=Homology_(mathematics)&oldid=1170405942
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can be any commutative ring (maybe even some noncommutative rings? I don't know).
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A finite disc including its boundary is (geometrically) closed but also bounded.
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By "subcomplex" above, I meant a subcomplex of a simplicial complex. That is,
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edit I therefore restored the original and asked for discussion here. However
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It's a good idea. I added an examples section inspired by the Italian page. --
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2646:{\displaystyle \ker(\partial _{0})/\mathrm {i} m(\partial _{1})=\mathbb {Z} }
1421:, and the foundation of the present Confusapedia article. (See Richeson, D.;
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What about the homology of product spaces? Doesn't that deserve a mention? --
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orientations. Hours after I edited the picture, I edited the referring text
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The Season's Greetings to you. Those manifolds were added to the table by
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Now taken a first pass. Anything in particular still missing? â Cheers,
1869:-module to start with instead of a free abelian group, nothing else.
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to see how the abstraction in this article could be somewhat tamed. --
2150:
more detailed treatment out to another more general article, such as
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Any coefficient group or ring we please can be used. So we can have
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Euler's Gem, The Polyhedron Formula and the Birth of Modern Topoplogy
534:
For a different perspective, compare this article to the articles on
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Shouldn't the article mention that homology theories satisfying the
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mentions the possibility of using coefficients in an arbitrary ring
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A boundary is a cycle which is also the boundary of a submanifold.
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a member of a different group or does that depend on the value of
1638:. Perhaps the phenomena that interest you are intrinsically mod
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and don't attempt to merge it in anywhere myself. The article on
265:, such that the composition of any two consecutive maps is zero:
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the same as the homology with torsion coefficients. If there's
756:
is it not mentioned anywhere? Where should it be covered? Would
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The drawing of cycles on the (hemispherical) projective plane
38:
1423:
Euler's Gem: The Polyhedron Formula and the Birth of Topology
211:. A chain complex is a sequence of abelian groups or modules
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over our editorial opinions, would be useful too. â Cheers,
2062:
What do you think? I hope I didn't put in too much jargon.
1566:
represent clockwise paths around 1 and 0, then the loop is
1377:
to welcome people in - to intimidate, not to communicate.
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can be written as the direct sum of a number of copies of
2419:
Now titled as closed and unbounded. For what it's worth:
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is a simplicial complex which is a union of simplices of
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https://en.wikipedia.org/Homology_(mathematics)#Surfaces
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1602:
Homology Theory for Real Analytic and Semianalytic Sets
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with a section on topological applications, or perhaps
794:, which would include the case of torsion coefficients
1740:â this turns up when studying differential equations).
1397:
as a way of investigating and classifying topological
869:
already means something else in materials science. --
825:
the resulting homology groups are finitely generated.
688:
Italian page should be used to expand the english page
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is that it is double-wound on the same line, just as
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used the idea of gluing and cutting, that was before
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thanks, this is clearer now. happy new year to you!
135:, a collaborative effort to improve the coverage of
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821:As far as I understand it (I'm not an expert), in
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197:The procedure works as follows: Given the object
360:. Is this correct? Or, to rephrase my question,
2965:Knowledge level-4 vital articles in Mathematics
8:
1225:-torsion in the integral homology in degree
921:, the existence of a short exact sequence:
833:, giving the Betti number, summed with the
364:does a chain complex encode information in
1710:(which might be zero), or what? â Cheers,
1218:{\displaystyle \mathbf {Z} /m\mathbf {Z} }
1179:{\displaystyle \mathbf {Z} /m\mathbf {Z} }
1144:{\displaystyle \mathbf {Z} /m\mathbf {Z} }
81:
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2755:Drawing of cycles on the projective plane
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2327:in part from Greek á˝ÎźĎĎ homos "identical"
1925:). Though in retrospect, I really meant
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579:Section "Construction of homology groups"
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2271:Four ways of gluing the square S2 error?
1880:is a subcomplex of a simplicial complex
1340:Shmuel (Seymour J.) Metz Username:Chatul
865:might be the best new article title, as
2955:Knowledge vital articles in Mathematics
494:Use for Homology? Example Calculations?
83:
42:
1490:willing to listen and help. â Cheers,
731:Betti numbers and torsion coefficients
430:I'm not sure but I think it should be
16:I removed this from the introduction:
2970:C-Class vital articles in Mathematics
2656:Does anyone have the missing pieces?
2534:{\displaystyle H_{0}(X)=\mathbb {Z} }
1642:, or perhaps it suffices to look mod
1527:-dimensional circles or loops) which
7:
1509:of intuitions may even be circular.
1151:. Additionally, it will retain the
129:This article is within the scope of
72:It is of interest to the following
2980:High-priority mathematics articles
2623:
2612:
2594:
2028:introduction to general relativity
1791:
1743:Geometrically, you can understand
1732:coefficients (the rational case),
1577:(the fundamental group is free on
1255:in the homology with coefficients.
514:
457:for the cohomology groups and not
14:
827:Finitely generated abelian groups
149:Knowledge:WikiProject Mathematics
2950:Knowledge level-4 vital articles
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1409:. In essence, it classifies the
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152:Template:WikiProject Mathematics
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2195:-manifold, for instance, is an
1809:{\displaystyle \partial ^{2}=0}
739:, Princeton University, 2008),
527:-complexes, see Allen Hatcher,
207:that encodes information about
169:This article has been rated as
2960:C-Class level-4 vital articles
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2376:orientability of disk and ball
1573:. This is non-trivial in the
1292:Torsion coefficient (topology)
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863:Torsion coefficient (homology)
766:Torsion coefficient (homology)
762:Torsion coefficient (topology)
758:Torsion coefficient (geometry)
296:+1-th map is contained in the
1:
2668:18:49, 26 February 2022 (UTC)
2455:13:22, 26 December 2021 (UTC)
2440:10:37, 25 December 2021 (UTC)
2415:10:24, 25 December 2021 (UTC)
2391:09:43, 25 December 2021 (UTC)
2322:08:13, 20 February 2018 (UTC)
2257:15:36, 2 September 2017 (UTC)
1830:21:45, 13 December 2015 (UTC)
1720:19:02, 13 December 2015 (UTC)
1673:17:44, 13 December 2015 (UTC)
1610:universal coefficient theorem
1550:15:50, 13 December 2015 (UTC)
1519:14:53, 13 December 2015 (UTC)
1251:, then it shows up in degree
890:universal coefficient theorem
837:, which is the direct sum of
426:Notation of cohomology groups
143:and see a list of open tasks.
2975:C-Class mathematics articles
2243:17:46, 19 January 2016 (UTC)
2228:14:05, 19 January 2016 (UTC)
2209:13:49, 19 January 2016 (UTC)
2187:08:34, 19 January 2016 (UTC)
2173:07:08, 19 January 2016 (UTC)
2145:21:38, 18 January 2016 (UTC)
2122:18:11, 18 January 2016 (UTC)
2072:15:18, 18 January 2016 (UTC)
2054:18:37, 13 January 2016 (UTC)
2040:13:25, 13 January 2016 (UTC)
2018:14:40, 12 January 2016 (UTC)
1951:13:44, 12 January 2016 (UTC)
1917:is continuous, I just meant
1903:a topological space, and if
1899:is a simplicial complex and
1845:12:33, 12 January 2016 (UTC)
1500:11:01, 2 December 2015 (UTC)
1482:00:41, 2 December 2015 (UTC)
1464:12:02, 1 December 2015 (UTC)
1450:10:50, 1 December 2015 (UTC)
1435:10:46, 1 December 2015 (UTC)
1387:04:21, 1 December 2015 (UTC)
1338:are canonically isomorphic?
1318:14:28, 14 January 2015 (UTC)
1304:16:57, 13 January 2015 (UTC)
1269:02:22, 13 January 2015 (UTC)
879:22:25, 12 January 2015 (UTC)
815:14:31, 12 January 2015 (UTC)
782:19:40, 11 January 2015 (UTC)
649:editing construction section
624:14:07, 29 October 2007 (UTC)
609:13:35, 29 October 2007 (UTC)
419:03:09, 28 October 2010 (UTC)
381:22:38, 21 October 2006 (UTC)
2928:08:44, 15 August 2023 (UTC)
2885:04:20, 15 August 2023 (UTC)
2866:03:00, 15 August 2023 (UTC)
2845:16:25, 14 August 2023 (UTC)
2786:has recently been changed.
2736:Correction made. â Cheers,
2464:In 'Informal examples' let
2218:ought to be possible here.
304:-th, and we can define the
2996:
2460:How do we get from X to C?
1348:21:44, 31 March 2015 (UTC)
725:23:30, 18 March 2014 (UTC)
2746:10:25, 25 June 2023 (UTC)
2732:08:38, 25 June 2023 (UTC)
2713:17:11, 16 July 2022 (UTC)
2697:15:42, 16 July 2022 (UTC)
2370:20:23, 3 April 2021 (UTC)
2341:21:17, 1 March 2021 (UTC)
1336:EilenbergâSteenrod axioms
1330:EilenbergâSteenrod axioms
710:21:05, 18 July 2012 (UTC)
683:15:26, 17 June 2012 (UTC)
663:04:30, 17 July 2010 (UTC)
644:04:05, 31 July 2008 (UTC)
574:04:29, 17 July 2010 (UTC)
560:00:37, 7 March 2007 (UTC
547:17:22, 9 April 2006 (UTC)
489:15:42, 2 April 2006 (UTC)
192:Request for clarification
168:
101:
80:
2306:07:27, 20 May 2016 (UTC)
2156:Introduction to topology
1753:to mean "cut along both
1598:EilenbergâMac Lane space
770:Introduction to homology
551:
372:13:43, 8 Apr 2005 (UTC)
175:project's priority scale
36:23:01, 12 Jun 2004 (UTC)
2490:{\displaystyle X=S^{1}}
1935:monad (category theory)
1694:intersect, then is say
892:: It asserts, for each
735:According to Richeson (
694:it:Omologia (topologia)
520:{\displaystyle \Delta }
507:simplicial (or perhaps
132:WikiProject Mathematics
2945:C-Class vital articles
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1983:
1865:means you used a free
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752:) but then I thought,
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309:-th homology group of
288:. This means that the
201:, one first defines a
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2779:{\displaystyle P^{2}}
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2568:
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2447:une musque de Biscaye
2383:une musque de Biscaye
1977:
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912:{\displaystyle i: -->
668:Homology and Homotopy
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477:{\displaystyle H_{n}}
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450:{\displaystyle H^{n}}
59:level-4 vital article
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2566:{\displaystyle C(X)}
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1658:-torsion and to mod
1407:torsion coefficients
1308:Now done. â Cheers,
1229:
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511:
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155:mathematics articles
1853:by taking the free
1401:according to their
1244:{\displaystyle i-1}
867:torsion coefficient
858:simplicial homology
823:simplicial homology
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2673:4 missing surfaces
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2215:algebraic K-theory
2024:general relativity
1984:
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1537:algebraic topology
1419:algebraic topology
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909:
692:The italian page,
529:Algebraic Topology
517:
474:
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318:(or factor module)
124:Mathematics portal
68:content assessment
2356:comment added by
1575:fundamental group
1353:The usual problem
788:Singular homology
616:singular homology
607:
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405:comment added by
232:... connected by
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2476:
2473:
2461:
2458:
2443:
2442:
2428:
2427:
2426:
2423:
2417:
2377:
2374:
2347:
2344:
2328:
2325:
2313:
2310:
2309:
2308:
2272:
2269:
2268:
2267:
2266:
2265:
2264:
2263:
2262:
2261:
2260:
2259:
2211:
2175:
2133:
2129:
2101:
2100:
2099:
2098:
2097:
2096:
2095:
2094:
2093:
2092:
2091:
2090:
2089:
2088:
2087:
2086:
2085:
2084:
2083:
2082:
2081:
2080:
2079:
2078:
2077:
2076:
2075:
2074:
2060:
1972:
1971:
1970:
1969:
1968:
1967:
1966:
1965:
1964:
1963:
1962:
1961:
1960:
1959:
1958:
1957:
1956:
1955:
1954:
1953:
1938:
1930:
1893:
1874:
1817:
1805:
1802:
1797:
1793:
1780:
1741:
1728:coefficients,
1633:
1620:
1605:
1594:
1532:
1511:193.91.226.149
1506:
1502:
1469:
1437:
1395:Henri PoincarĂŠ
1379:110.20.158.134
1354:
1351:
1331:
1328:
1327:
1326:
1325:
1324:
1323:
1322:
1321:
1320:
1274:
1273:
1272:
1271:
1256:
1240:
1237:
1234:
1213:
1209:
1205:
1200:
1174:
1170:
1166:
1161:
1139:
1135:
1131:
1126:
1114:
1113:
1112:
1101:
1098:
1095:
1091:
1087:
1083:
1078:
1074:
1071:
1067:
1063:
1060:
1057:
1052:
1049:
1046:
1042:
1038:
1035:
1032:
1029:
1026:
1022:
1018:
1014:
1009:
1005:
1002:
999:
994:
990:
986:
982:
978:
974:
969:
965:
962:
958:
954:
951:
948:
943:
939:
935:
932:
908:
905:
902:
882:
881:
818:
817:
732:
729:
728:
727:
689:
686:
669:
666:
650:
647:
631:
628:
627:
626:
580:
577:
553:
550:
516:
495:
492:
471:
467:
444:
440:
427:
424:
395:
391:
387:
384:
383:
354:
353:
346:
337:
324:
319:
278:
269:
259:
255:
248:
239:
229:
222:
215:
193:
190:
187:
186:
183:
182:
179:
178:
167:
161:
160:
158:
141:the discussion
128:
127:
111:
99:
98:
90:
78:
77:
71:
49:
26:
25:
13:
10:
9:
6:
4:
3:
2:
2992:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2942:
2940:
2929:
2925:
2921:
2917:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2887:
2886:
2882:
2878:
2873:
2869:
2867:
2863:
2859:
2854:
2849:
2848:
2847:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2808:
2796:
2787:
2771:
2767:
2754:
2747:
2743:
2739:
2735:
2733:
2729:
2725:
2720:
2719:
2718:
2717:
2714:
2710:
2706:
2701:
2700:
2699:
2698:
2694:
2690:
2686:
2683:
2682:
2678:
2672:
2670:
2669:
2665:
2661:
2657:
2654:
2635:
2627:
2616:
2607:
2598:
2587:
2584:
2557:
2551:
2542:
2523:
2517:
2509:
2505:
2482:
2478:
2474:
2471:
2459:
2457:
2456:
2452:
2448:
2441:
2437:
2433:
2429:
2424:
2421:
2420:
2418:
2416:
2412:
2408:
2403:
2399:
2395:
2394:
2393:
2392:
2388:
2384:
2375:
2373:
2371:
2367:
2363:
2359:
2355:
2345:
2343:
2342:
2338:
2334:
2326:
2324:
2323:
2320:
2311:
2307:
2303:
2299:
2294:
2293:
2292:
2289:
2285:
2281:
2277:
2270:
2258:
2254:
2250:
2246:
2245:
2244:
2240:
2236:
2231:
2230:
2229:
2225:
2221:
2216:
2212:
2210:
2206:
2202:
2198:
2194:
2190:
2189:
2188:
2184:
2180:
2176:
2174:
2170:
2166:
2161:
2157:
2153:
2148:
2147:
2146:
2142:
2138:
2134:
2130:
2126:
2125:
2124:
2123:
2119:
2115:
2110:
2106:
2073:
2069:
2065:
2061:
2057:
2056:
2055:
2051:
2047:
2043:
2042:
2041:
2037:
2033:
2029:
2025:
2021:
2020:
2019:
2015:
2011:
2006:
2005:
2004:
2003:
2002:
2001:
2000:
1999:
1998:
1997:
1996:
1995:
1994:
1993:
1992:
1991:
1990:
1989:
1988:
1987:
1986:
1985:
1981:
1976:
1952:
1948:
1944:
1939:
1936:
1931:
1928:
1924:
1920:
1915:
1911:
1907:
1902:
1898:
1894:
1891:
1887:
1883:
1879:
1875:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1847:
1846:
1842:
1838:
1833:
1832:
1831:
1827:
1823:
1818:
1803:
1800:
1795:
1781:
1778:
1774:
1769:
1765:
1760:
1756:
1751:
1747:
1742:
1739:
1735:
1731:
1727:
1723:
1722:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1676:
1675:
1674:
1670:
1666:
1661:
1657:
1653:
1649:
1645:
1641:
1636:
1632:
1628:
1623:
1619:
1615:
1611:
1606:
1603:
1599:
1595:
1592:
1588:
1584:
1580:
1576:
1572:
1569:
1565:
1561:
1557:
1553:
1552:
1551:
1547:
1543:
1538:
1533:
1530:
1526:
1522:
1521:
1520:
1516:
1512:
1507:
1503:
1501:
1497:
1493:
1489:
1485:
1484:
1483:
1479:
1475:
1470:
1467:
1466:
1465:
1461:
1457:
1453:
1452:
1451:
1447:
1443:
1438:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1403:Betti numbers
1400:
1396:
1391:
1390:
1389:
1388:
1384:
1380:
1374:
1370:
1366:
1362:
1358:
1352:
1350:
1349:
1345:
1341:
1337:
1329:
1319:
1315:
1311:
1307:
1306:
1305:
1301:
1297:
1293:
1289:
1288:Betti numbers
1285:
1280:
1279:
1278:
1277:
1276:
1275:
1270:
1266:
1262:
1257:
1254:
1238:
1235:
1232:
1207:
1203:
1189:
1168:
1164:
1133:
1129:
1115:
1099:
1085:
1081:
1072:
1061:
1058:
1050:
1047:
1044:
1040:
1033:
1030:
1016:
1012:
1003:
1000:
992:
988:
976:
972:
963:
952:
949:
941:
937:
930:
923:
922:
906:
903:
900:
891:
886:
885:
884:
883:
880:
876:
872:
868:
864:
859:
855:
851:
848:
844:
840:
839:cyclic groups
836:
832:
828:
824:
820:
819:
816:
812:
808:
804:
801:
797:
793:
789:
786:
785:
784:
783:
779:
775:
771:
767:
763:
759:
755:
751:
747:
746:Betti numbers
742:
741:Betti numbers
738:
730:
726:
722:
718:
714:
713:
712:
711:
707:
703:
698:
695:
687:
685:
684:
680:
676:
667:
665:
664:
660:
656:
648:
646:
645:
641:
637:
629:
625:
622:
617:
613:
612:
611:
610:
605:
601:
596:
590:
589:
585:
578:
576:
575:
571:
567:
561:
559:
549:
548:
545:
544:Michael Stone
541:
537:
536:covering maps
532:
530:
504:
500:
493:
491:
490:
487:
469:
465:
442:
438:
425:
423:
420:
416:
412:
408:
404:
382:
379:
375:
374:
373:
371:
367:
363:
359:
349:
345:
340:
336:
332:
327:
323:
320:
317:
313:
312:
308:
303:
299:
295:
291:
287:
281:
277:
272:
268:
262:
258:
253:
251:
247:
242:
238:
235:
234:homomorphisms
228:
221:
214:
210:
206:
205:
204:chain complex
200:
196:
195:
191:
176:
172:
171:High-priority
166:
163:
162:
159:
142:
138:
134:
133:
125:
119:
114:
112:
109:
105:
104:
100:
96:Highâpriority
94:
91:
88:
84:
79:
75:
69:
61:
60:
50:
46:
41:
40:
37:
35:
30:
23:
19:
18:
17:
2898:
2894:
2890:
2820:
2818:
2758:
2687:
2684:
2679:
2676:
2658:
2655:
2543:
2463:
2444:
2379:
2352:â Preceding
2349:
2330:
2315:
2290:
2286:
2282:
2278:
2274:
2196:
2192:
2159:
2108:
2104:
2102:
1980:Klein bottle
1978:Cycles on a
1926:
1922:
1918:
1913:
1909:
1905:
1900:
1896:
1889:
1885:
1881:
1877:
1870:
1866:
1862:
1858:
1854:
1850:
1776:
1772:
1767:
1763:
1758:
1754:
1749:
1745:
1738:local system
1733:
1729:
1725:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1659:
1655:
1651:
1647:
1643:
1639:
1634:
1630:
1626:
1621:
1617:
1613:
1601:
1590:
1586:
1582:
1578:
1570:
1567:
1563:
1559:
1528:
1524:
1487:
1422:
1415:Emmy Noether
1410:
1375:
1371:
1367:
1363:
1359:
1356:
1333:
1294:. â Cheers,
1252:
1187:
849:
846:
842:
841:of the form
830:
802:
799:
795:
791:
772:? â Cheers,
753:
736:
734:
699:
691:
675:70.247.166.5
671:
652:
633:
591:
587:
586:
582:
562:
558:149.4.108.33
555:
533:
505:
501:
497:
429:
398:(X)â mean?
385:
365:
361:
357:
355:
347:
343:
338:
334:
330:
325:
321:
316:factor group
310:
306:
305:
301:
293:
285:
284:= 0 for all
279:
275:
270:
266:
260:
256:
249:
245:
240:
236:
226:
219:
212:
208:
202:
198:
170:
130:
74:WikiProjects
57:
34:Toby Bartels
31:
27:
15:
2920:Steelpillow
2837:Steelpillow
2831:has chosen
2812:New version
2738:Steelpillow
2724:Steelpillow
2705:Steelpillow
2432:Steelpillow
2407:Steelpillow
2398:Rockyunited
2298:Steelpillow
2249:Woscafrench
2235:Steelpillow
2179:Steelpillow
2165:Steelpillow
2114:Steelpillow
2046:Steelpillow
2010:Steelpillow
1837:Steelpillow
1712:Steelpillow
1542:Steelpillow
1492:Steelpillow
1456:Steelpillow
1442:Steelpillow
1427:Steelpillow
1310:Steelpillow
1296:Steelpillow
871:Mark viking
774:Steelpillow
717:Mark viking
702:JackSchmidt
401:âPreceding
390:(X)â and âB
378:Orthografer
146:Mathematics
137:mathematics
93:Mathematics
2939:Categories
2902:consensus.
2577:, nor how
2430:â Cheers,
919:0}" /: -->
655:Particle25
630:Properties
566:Particle25
540:homotopies
314:to be the
2689:Darcourse
2660:Darcourse
2402:this edit
1399:manifolds
636:Raijinili
564:compute.
407:NormHardy
62:is rated
2914:putting
2911:homotopy
2907:topology
2877:Chaikens
2858:Chaikens
2829:Chaikens
2405:Cheers,
2366:contribs
2358:Commevsp
2354:unsigned
2160:homology
2152:Topology
2109:homology
2105:topology
1646:for all
1625:and the
1556:homotopy
1284:my draft
896:0}": -->
604:Contribs
415:contribs
403:unsigned
333:) = ker(
22:manifold
2497:, then
2319:Lambiam
621:Zundark
486:Cheesus
342:) / im(
300:of the
292:of the
173:on the
64:C-class
2154:or an
1411:cycles
595:Maelin
298:kernel
254:-: -->
70:scale.
2916:WP:RS
904:: -->
370:Lupin
290:image
51:This
2924:Talk
2897:and
2881:talk
2862:talk
2841:Talk
2833:here
2825:this
2742:Talk
2728:Talk
2709:Talk
2693:talk
2664:talk
2451:talk
2436:Talk
2411:Talk
2387:talk
2362:talk
2337:talk
2302:Talk
2253:talk
2239:Talk
2224:talk
2220:Ozob
2205:talk
2201:Ozob
2183:Talk
2169:Talk
2141:talk
2137:Ozob
2118:Talk
2068:talk
2064:Ozob
2050:Talk
2036:talk
2032:Ozob
2026:and
2014:Talk
1947:talk
1943:Ozob
1841:Talk
1826:talk
1822:Ozob
1775:and
1757:and
1716:Talk
1690:and
1682:and
1669:talk
1665:Ozob
1589:and
1581:and
1562:and
1546:Talk
1515:talk
1496:Talk
1478:talk
1474:Ozob
1460:Talk
1446:Talk
1431:Talk
1405:and
1383:talk
1344:talk
1314:Talk
1300:Talk
1265:talk
1261:Ozob
875:talk
811:talk
807:Ozob
778:Talk
750:here
721:talk
706:talk
679:talk
659:talk
640:talk
600:Talk
570:talk
538:and
411:talk
165:High
2677:In
2585:ker
2400:in
1884:if
1654:to
1568:xyx
1529:can
1488:are
1188:not
1031:Tor
764:or
754:why
484:--
362:how
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2941::
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813:)
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413:â˘
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352:).
350:+1
282:+1
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263:-1
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225:,
218:,
2922:(
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2891:b
2879:(
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2748:]
2740:(
2726:(
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2633:)
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2524:=
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2510:0
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2449:(
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2360:(
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1927:f
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1914:X
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1890:C
1886:D
1882:C
1878:D
1871:R
1867:R
1863:R
1859:R
1855:R
1851:R
1839:(
1824:(
1804:0
1801:=
1796:2
1777:b
1773:a
1768:b
1764:a
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1755:a
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1746:a
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1730:Q
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1714:(
1708:b
1704:a
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1696:a
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1660:p
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1627:p
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1614:p
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1066:Z
1062:;
1059:X
1056:(
1051:1
1045:i
1041:H
1037:(
1025:)
1021:Z
1017:m
1013:/
1008:Z
1004:;
1001:X
998:(
993:i
989:H
981:Z
977:m
973:/
968:Z
961:)
957:Z
953:;
950:X
947:(
942:i
938:H
931:0
907:0
901:i
873:(
850:Z
847:m
845:/
843:Z
831:Z
809:(
803:Z
800:m
798:/
796:Z
792:R
776:(
719:(
704:(
677:(
657:(
638:(
606:)
598:(
568:(
470:n
466:H
443:n
439:H
409:(
396:n
392:n
388:n
366:X
358:X
348:n
344:d
339:n
335:d
331:X
329:(
326:n
322:H
311:X
307:n
302:n
294:n
286:n
280:n
276:d
271:n
267:d
261:n
257:A
250:n
246:A
241:n
237:d
230:2
227:A
223:1
220:A
216:0
213:A
209:X
199:X
177:.
76::
24:.
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