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 Field: Topology

Correct Definition?[edit]

Is the definition set up correctly for homotopy equivalence? It seems if you have a disk, X, and a point, Y = {y}, there is only one function f:X->Y, f(x) = y. Take a, b in X such that a /= b. Thus, g(f(a)) = g(y) = g(f(b)), so if g*f is the identity function, we have a contradiction that a = b. (talk) 14:03, 28 July 2008 (UTC)

We only require the functions f(g) and g(f) to be homotopic to the identity. In your case, a constant function from a disk to itself is homotopic to the identity function and any function from a point to itself is the identity. Orthografer (talk) 02:22, 31 July 2008 (UTC)

And actually a topological space is call contractable if (and only if) it is homotopic to a point. Dharma6662000 (talk) 17:37, 25 August 2008 (UTC)
Continuous functions have homotopies, but not topological spaces. Orthografer (talk) 14:20, 26 August 2008 (UTC)

When is H continuous?[edit]

What topology do we usually assume on functions X->Y? The definition in the article says H must be continuous, but it is not explained when H is continuous. (talk) 14:03, 28 July 2008 (UTC)

You are asking two nice questions here. I think that one of the questions you don't realise your asking. So first of all I'll answer the question I think you're asking, i.e. "What hoes it mean for to be continuous</math>. Well Y is a topolgical space so we have a collection of open sets which cover Y and satisfy some axioms. For H to be continuous we require that the preimage of every open set of Y is open in . For the open sets in we consider the product topology given by the topology on X and the topology on . The second question which I don't think you meant to ask was "What is a topology on the space of maps ?" This is a very nice question since this space is an infinite dimensional space. If we restrict our attention to the space of smooth functions, i.e. then we can use the Whitney -topology. This compares k-jets and successive partial derivatives to give a notion of proximity it this infinite dimensional space. This is a very interesting question, and I urge you to study it in depth. Dharma6662000 (talk) 17:57, 25 August 2008 (UTC)
A simple answer is to give a topology in Xx[0,1] and then H:Xx[0,1]->Y is continuous if for each open set U in Y, we have H^(-1)U is open in Xx[0,1], that easy--kmath (talk) 19:59, 25 August 2008 (UTC)
That's what my rely said (and I quote: "For H to be continuous we require that the preimage of every open set of Y is open in . "), why bother to say the same thing twice? The key problem is what are the open sets on  ? And my reply answers that question too by saying that the product topology should be used. Dharma6662000 (talk) 20:28, 25 August 2008 (UTC)
Now our friend is going to be focused on that matter--kmath (talk) 02:10, 26 August 2008 (UTC)

Incorrect statment?[edit]

I think it is not true as stated in the article, that when X and Y are homotopy equivalent, and X is locally path connected, then Y is also locally path connected. For an example, take the subspace of a unit square consisting of the interval [(0,0),(0,1)] together with all line segments from (0,1) to the points (1/n,0) where n runs through all natural numbers. The space is contractible, but it is not locally path connected at (0,0).

I am not sure that the statement that two homotopies can be rotated makes much sense (but I know what is meant..:) )

Yes, locally path-connected isn't a homotopy invariant.

It seems to me the stuff on homotopy groups belongs in its own article. --gorlim 16:52, 24 Apr 2004 (UTC)

The stuff on homotopy groups has its own page. It seems to me that the stuff on isotopy belongs in its own article. —Blotwell 08:40, 23 July 2005 (UTC)

Misleading picture?[edit]

The illustration with caption "An illustration of a homotopy between the two bold paths" is somewhat confusing for two reasons:

1. first, which subsets of the bold loop correspond to the "two bold paths" could be misinterpreted, making the isolines even more confusing

2. second, at first glance, the illustration makes you think that all homotopies are necessarily relative to some none-empty subset, since the endpoints of the curves are fixed.

I'm changing the caption for reason #2, feel free to change it back...

I think the etimology here is wrong: homo-topy comes from the greek verb "topao"(= to melt, or to deform), and not from the name "topos" (= place).

(in reference to above) was the etymology clarified? 00:29, 22 June 2007 (UTC)

From the OED: ORIGIN from HOMO + -Greek topos (place) + -y —Preceding unsigned comment added by (talk) 11:17, 8 January 2009 (UTC)

Why are all the illustrations isotopies?[edit]

Would be nice if someone would create/add an illustration of a homotopy that's not an isotopy :)

Homeomorphism Homotopy[edit]

Maybe a note should be added that says that all homeomorphisms are homotopies. (Or have I missed it?) Two continuous maps are called homotopic if (and only if) there exists a continuous map such that and for all . Now, two topological spaces X and Y are homotopic if (and only if) there exist continuous maps and such that and are both homotopic to the identity maps. Now, assume that X and Y are homeomorphic, then we can choose a homeomorphism and let . Then and . It follows that X and Y are homotopic. Dharma6662000 (talk) 17:46, 25 August 2008 (UTC)

I think you're imagining these topological spaces as embedded in other spaces. A homeomorphism is not necessarily a homotopy.Orthografer (talk) 14:22, 26 August 2008 (UTC)
What's wrong with the above proof? It makes no mention of an ambient space and deals only with abstract topological spaces. The proof seems correct to me. The proof shows that two spaces that are homeomorphic are necessarily homotpic. See pages 5 and 6 of
  • H. Osborn's Vector Bundles, Volume 1: Foundations and Stiefel-Whitney Classes. Academic Press 1982. Declan Davis (talk) 14:53, 26 August 2008 (UTC)
In fact the article says that all homeomorphisms are homotopies. In homotopy equivalence and null-homotopy the article says: "Clearly, every homeomorphism is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent." Declan Davis (talk) 16:01, 26 August 2008 (UTC)
I agree that a homeomorphism is a homotopy equivalence. My point is the following: saying that a homeomorphism is a homotopy is, strictly speaking, not true. Further, it distracts the reader from the main idea, which is that a homotopy is simply a 1-parameter family of maps. Certainly a trivial homotopy can be constructed from any continuous map. What is special about such a map being a homeomorphism? I also think some of the people here are confusing the word "homotopic" with "deformation equivalent." Orthografer (talk) 18:43, 24 October 2008 (UTC)
I completely agree with Orthografer. There is a clear confusion supra: an homotopy equivalence is not an homotopy. I think this should be clarified explicitly in the article.Moreover, I'm a bit uncomfortable with the coffey cup <---> doughnut illustration; it does illustrate an homotopy (and indeed even an isotopy), provided that the cup and the torus are considered not as topological spaces, but as two different embeddings of the same space into (e.g.) R3. We should not be too formal; but also not incorrect. JoergenB (talk) 20:38, 31 October 2008 (UTC)

I clarified the coffee mug example in the text of the section, and in the caption to the animation. MorphismOfDoom (talk) 17:58, 19 August 2014 (UTC)

Homeomorphisms per se have nothing to do with homotopies. A homeomorphism is a certain kind of mapping (continuous and bijective with continuous inverse) between topological spaces. A homotopy is a continuous family of mappings between two topological spaces. The existence of a homotopy between two continuous mappings X → Y defines an equivalence relation on those mappings. Some homeomorphisms between spaces X and Y (if there exist any at all!) are homotopic, some are not. Yes, all homeomorphisms are homotopy equivalences — but the concept of a homotopy equivalence is quite different from that of a homotopy.Daqu (talk) 06:42, 26 January 2017 (UTC)


Firstly, should the alternative definition of a homotopy from f to g, in terms of a family of maps ht such that h0= f and h1= g be included? Both versions are used so should both be in the article?

Secondly, why does it say that the fundamental group can only be defined with a concept of homotopies relative to a subspace? At most, all you need is relative to a point, and that is a far simpler concept. Saying relative homotopy is needed for the fundamental group makes it sound more complicated than it is. Zerxp —Preceding not properly signed comment added by Zerxp (talkcontribs) 13:51, 1 November 2008 (UTC)

Hi, and welcome, Zerxp!
The "alternative definition" is equivalent to the ordinary one. Thus, very formally, it is unneccessary; but I think it is good to include it, since it may help the reader to understands what goes on.
When you define homotopy relative to a point, that is equivalent to defining it relative to the subspace consisting of only that point. Thus, the subset definition formally includes the one point subset one. However, IMHO you are right in one sense, since the formulation with subsets is slightly more complicated, if you are only going to apply it for the ordinary homotopy groups anyhow. JoergenB (talk) 14:05, 3 November 2008 (UTC)
(PS. The easiest way to sign your discussion contributions "properly" is to push the "signature button", or to write ~~~~ by hand. Test it; its effects are "slightly magical"!)
Thanks for the help with signing, that will be useful.
I have put in the other definition, along with how it relates to the original one given, just to try to help avoid confusion on the matter for people who are new to homotopy. I have left the fundamental group as part of the Homotopy groups paragraph, but made sure to put a link to its own page and so that page can explain it without having to use the relative homotopy groups, but I decided that it would take up a lot of space on this homotopy page to include the construction of the fundamental group without using the relative homotopy that has already been described.
Also, I have added a small paragraph on the Homotopy lifting property as there was currently no link at all to that page, and I think it is another important result. I don't know why, but the size of some of the maths text has come out smaller than the rest, which makes it look very odd to me, but I do not know how to fix that. Zerxp (talk) 11:06, 6 November 2008 (UTC)

Alternative definition appears incorrect or incomplete[edit]

I object to the following as a definition of a homotopy:

An alternative notation is to say that a homotopy between two continuous functions f , g : XY is a family of continuous functions ht: XY for t ∈ [0,1] such that h0= f and h1= g and the map tht is continuous from [0,1] to the space of all continuous functions XY. The two versions coincide by setting ht(x) = H(x,t).

This is meaningless so long as no topology has been put on the set of continuous functions from X to Y.

The definition would be correct if this set were given the compact-open topology and X was locally compact. This is too complicated, though. I think the best thing to do is to say that the family ht is "continuous" only in the sense that the original definition is satisfied, where H(x,t) = ht(x). (talk) 12:45, 14 November 2008 (UTC)

I do not recall any mention of a topology on the function space Ω(X,Y) when I was learning about homotopies. The (usual) definition of homotopy says that two maps f : XY and g : XY are homotopic if there exists a continuous map H : X × [0,1] → Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. This definition only requires that we have a topology on the spaces X × [0,1] and Y. I've never seen any mention of the map [0,1] → Ω(X,Y) needing to be continuous. Maybe if we seek a special kind of homotopy then this extra condition may be imposed.  Δεκλαν Δαφισ   (talk)  22:42, 14 November 2008 (UTC)

cofee cup picture[edit]

Is the cofee cup picture appropriate in this article? It seems to be a full homeomorphism and not just an homotopy equivalence. If the torus were a solid torus and it became a circumference it would. The current one may give the impression that a homotopy equivalence is a stronger relation than it is. —Preceding unsigned comment added by (talk) 20:39, 20 May 2009 (UTC)

I also think the coffee cup picture is misleading, for a different reason. A homotopy is a continuous deformation between two functions, where each function is a continuous deformation between two topological spaces. But the coffee cup picture just shows one continuous deformation between two topological spaces. Maproom (talk) 20:52, 16 February 2011 (UTC)
Doesn't it show a parametrized family of embeddings of certain spaces into R3, starting with the cup and ending with a torus? — Carl (CBM · talk) 20:57, 16 February 2011 (UTC)
Yes, you are right. It's easy to look at that image and not notice the embedding R3. Maproom (talk)
The picture is good only for illustrating the Homeomorphism relation, but to be honest would be better if it was removed. —Preceding unsigned comment added by (talk) 21:38, 24 February 2011 (UTC)
The picture should be either explained better (indicating what are X, Y, f and g), or removed as confusing without further explanation. Now the reader may take away the incorrect impression that isotopy means a reversible continuous mapping, i.e., homeomorphism. It's easy enough to say X = S1 × S1 (a torus) and Y = R3. But how to define f and g without too much handwaving?  --Lambiam 22:31, 20 November 2012 (UTC)

I restored the coffee mug animation after initially having deleted it, added a clear explanation of what the homotopy is (what X, Y, f, and g are, and what the animated image shows) in the text at the end of the " formal definition " section, and made the caption accurate as well. MorphismOfDoom (talk) 17:52, 19 August 2014 (UTC)

Homology, homotopy and homeomorphism[edit]

I would find it really useful to have a dummies' explanation of the differences between all these things, including such subtleties as distinguishing homotopies vs. homotopy equivalents vs homeomorphisms. Examples would be good. At present the article reads little better a student's notes. — Cheers, Steelpillow (Talk) 22:29, 25 January 2011 (UTC)

Completely agree. Will try to work on this. - Subh83 (talk | contribs) 04:43, 22 November 2011 (UTC)

New article for homotopy equivalence[edit]

I would like to suggest a separate article for "Homotopy equivalence" since it is significantly more advanced than the notion of simple homotopy. Moreover it has applications in may different context that need special mention. For example, homology groups and homotopy groups are invariant up to homotopy equivalence. Moreover its relation with and distinction from deformation retract is worth mentioning.

Finally, and most importantly, a clear clarification of the difference between two spaces being "homotopic" vs. them being "homotopy equivalent" is important to avoid confusion.

If these make sense, I will start working on the article. - Subh83 (talk | contribs) 04:34, 22 November 2011 (UTC)

the isotopy part plenty of errors!!!![edit]

"and an embedding is simply a homeomorphism" WHAT?!!! they are completely different concepts!!!

"The intuitive idea of deforming one to the other should correspond to a path of embeddings: [...] this corresponds to the definition of isotopy"

no! this is not! The given definition of isotopy is about homeomorphisms, non embeddings!! -- (talk) 19:42, 13 January 2013 (UTC)

I think the one who wrote the sub-section used definitions more aligned along ambient isotopy (the article for which, itself, has many things incorrect in it).
I have completely re-worked the subsection to reflect the standard definition of isotopy.
Corrected the definition: "In case the two given continuous functions f and g from the topological space X to the topological space Y are embeddings, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x,t) gives a embedding."
Added clarification "...and this embedding gives a homeomorphism between the circle and its image in the embedding space...". (see Embedding#General_topology). Although I do not see why this statement is relevant here. It can very well be removed.
Fixed the sentences about knots: "The intuitive idea behind the notion of knot equivalence is that one can deform one embedding to another through a path of embeddings: a continuous function starting at t=0 giving the K1 embedding, ending at t=1 giving the K2 embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy."
Removed this sentence since isotopy has nothing to do with smoothness: "This works well as long as we require all the maps involved to be differentiable, but fails if they are simply allowed to be continuous, as there is then no obstruction to pulling the knot taut, which changes the knot geometry." If the knot is 'taut', it no more remains an embedding.
Not sure about this sentence: "An ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K1 and K2 are considered equivalent when there is an ambient isotopy which moves K1 to K2. This is the appropriate definition in the topological category. " Have kept it though.
Here is the diff.
- Subh83 (talk | contribs) 19:24, 31 January 2013 (UTC)

Why "Formal" Definition?[edit]

Why "Formal" Definition? Does that label help people understand the article? I find it distracting. (talk) 02:41, 10 November 2013 (UTC)

First of all, article is technical at times, it is fine. Wikipedia sole purpose is to document and universal readability is not our goal.
I also don't know why you find formal definition distracting but one can add its own section of (plain) definition.
Lastly, the lead section, In topology, two continuous functions from one topological space to another are called homotopic (Greek ὁμός (homós) = same, similar, and τόπος (tópos) = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
Feel free to tell me/us what is missing. -- (talk) 08:29, 18 December 2013 (UTC)

Guess what? Universal readability should be your goal. Maybe this explains why so many math articles in Wikipedia are really, really, really bad.Daqu (talk) 06:46, 26 January 2017 (UTC)


Does anyone know the etymology of the word "homotopy"? I would guess it comes from the greek word "topos"="place" such as "utopia" or "distopia", but it is only a guess. If someone could check the etymology of the word and add it to the page it would be much appreciated.

I don't know the details. But basically, Greek "homo" = same and "topos" = place. Maproom (talk) 21:29, 9 January 2014 (UTC)

Error in Homotopy Equivalence section[edit]

I believe the following is incorrect:

[F]or example, the double torus and the double torus with the rings interlinked are homotopy equivalent (since they are homeomorphic), even though the said transformation cannot be embedded in three-dimensional Euclidean space without the rings "passing through" each other.

This transformation may indeed be embedded in R^3. Here is an illustration of exactly that. I have edited this out for now. If someone knows of a better example, please include it.

Peligro~enwiki (talk) 01:56, 18 November 2015 (UTC)

Maybe a valid example is a torus and a knotted torus. Maproom (talk) 11:10, 18 November 2015 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Homotopy/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

At the moment this is the main article on homotopy theory. Geometry guy 19:52, 10 June 2007 (UTC)

Last edited at 19:52, 10 June 2007 (UTC). Substituted at 02:13, 5 May 2016 (UTC)

False caption to illustration[edit]

The caption to the illustration of an unknot and a trefoil knot reads as follows:

"The unknot is not equivalent to the trefoil knot since one cannot be deformed into the other through a continuous path of embeddings. Thus they are not ambient-isotopic."

The statement that "[O]ne cannot be deformed into the other through a continuous path of embeddings" is false. The intuitive idea of pulling the knot tight can be made rigorous so that at the final stage the knot disappears entirely, leaving an unknot.Daqu (talk) 17:47, 23 January 2017 (UTC)

Are you trying to say the trefoil is equivalent to the unknot? By pulling tight, do you mean shrinking to a single point, which would not be an embedding? — Carl (CBM · talk) 12:13, 26 January 2017 (UTC)
What I am saying is that a continuous family of merely topological embeddings of the circle into 3-space exists between any knot and the unknot. The intuitive idea is simply to pull the knotted part of the curve tight. Because the length of the knotted part of the curve approaches 0 in the limit as t → 1, this is a continuous family of embeddings.
For this reason, mere isotopy of embedded circles is not an adequate way of defining knot equivalence. (Ambient isotopy — an isotopy of the entire 3-space carrying one knot to another — is an adequate way of defining knot equivalence.)Daqu (talk) 00:46, 2 February 2017 (UTC)

Yet another reason why this article is bad[edit]

The animated illustration with the caption

"A homotopy between two embeddings of the torus into R3: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an isotopy."

is not an example of either a homotopy or an isotopy.

The reason this is not an example of either one is that in a homotopy or an isotopy, specific points move in specific paths as the "time" parameter increases from 0 to 1. But this illustration gives no indication of where specific points move. So although it illustrates something, that something is neither a homotopy nor an isotopy.Daqu (talk) 06:52, 26 January 2017 (UTC)

I think this is a little unfair. The illustration shows the embeddings obtained at various points of time along a homotopy. Since the homotopy itself is a function, with a graph in a high-dimensional space, it seems impossible to sketch the actual homotopy function. But it is common to illustrate a homotopy by showing various stages, without tracking particular points. — Carl (CBM · talk) 12:15, 26 January 2017 (UTC)
I am complaining about a false statement. The illustration makes no attempt to indicate that the isotopy of a solid torus between the bagel shape and the coffee-mug shape is actually a motion of each point of one embedding, along a path, to a specific point of the other embedding.
The statement "The illustration shows the embeddings obtained at various points of time along a homotopy" is not true. The illustration shows only the images of the embeddings obtained at various points of time along a homotopy (that is actually an isotopy, a very special kind of homotopy).
The important thing is that anyone coming to this article with the goal of understanding what homotopy means will very likely be confused.Daqu (talk) 00:37, 2 February 2017 (UTC)