Talk:Retraction (topology)

Hmmmm...

"a continuous map r is a retract if ... In this case, A is called a retract of X, and r is called a retraction"

make up your mind, dudes

Fixed. Apart from the repetition, the map is called a retraction, while the subspace is a retract. There was some confusion about it. Alvisetrevi 10:25, 27 September 2007 (UTC)

strong deformation retract?

I've seen the phrase "strong deformation retract". How's that related to "deformation retract"? thanks! --345Kai 16:10, 5 March 2007 (UTC)

Definition

According to the history, people seem to keep changing the third line of the definition from d(a,t)=a (correct) to d(a,1)=a (true but incorrect for the definition). Most sources (eg. Hatcher) seem to favour the first definition, but more importantly, the second one is inconsistent with the claim at the end that "contractible spaces exist which do not deformation retract to a point", and also makes defining t redundant. Please stop changing it back - it stumped me for a good 15 minutes and I'm sure it'll stump other assignment-doers too =P Pirsq 11:58, 22 June 2007 (UTC)

Well, I don't think that choosing a definition according to whether or not it contrasts with other claims in a Wikipedia page is a good idea... suppose we choose to define a deformation retract using d(a,1)=a and a strong deformation retract using d(a,t)=a. In this case we could modify the claim to "contractible spaces exist which do not strongly deformation retract to a point" (or something like that) and we're all set! The definition with d(a,1)=a has the advantage, in my opinion, of being equivalent to the following concise and elegant definition:

Let $\iota :A\hookrightarrow X$ be the inclusion. A 'deformation retract $r$ is a retract such that $\iota \circ r$ is homotopic to the identity of X.

This has a one-line proof! Another advantage is that the homotopy equivalence of A and X is immediate. If no one disagrees within a couple of days I will edit the entry.Alvisetrevi 20:34, 26 September 2007 (UTC)

Then people seem to be using the following definitions: strong deformation retract, deformation retract. In most sources I have seen these two definitions are used, but haven't checked enough of them to argue about sources in general. —Preceding unsigned comment added by 83.9.83.147 (talk) 18:52, 25 September 2007 (UTC)

Sorry, I just caught up with my watchlist and saw this. After checking four or five of the "standard" textbooks, it appears that there is disagreement about definitions. The ideal article would not only mention the prevailing definitions, citing the textbooks that use them, but would also address the issue of which definitions are needed for which results. (Certain statements are only true using the "stronger" form of deformation, for example.) It might make for a little more bulk than we would want, but we can't just sweep this issue under the rug. Now that Hatcher is fast becoming the go-to textbook, I want to be especially careful that we don't ignore the definitions there, which differ from the ones here. I am swamped for the moment, but I'll try to come back to this if I get time. VectorPosse 16:40, 1 October 2007 (UTC)

Yes, what you say definitely makes sense. Of course that needs time, and time is money!. What follows is just a "bibliographical" note. From my memory, Hatcher and May (A concise course in AT) use the stronger version, while Bredon, Bott&Tu and Rotman (An intro to AT) use the weaker one. I just checked Massey: he goes for the stronger one, but mentions the existence of different uses. Alvisetrevi 08:38, 2 October 2007 (UTC)

Example?

"However, there exist contractible spaces which do not strongly deformation retract to a point." An example of this here would be fantastic. msgj 09:18, 11 October 2007 (UTC)

I've been puzzeling over this for some time and an example is the line with two origins. It can be retracted to the top origin, but during the courser of this deformation the bottom origin needs to leave the origin to get to the top origin, on the way it has to take the top origin with it, thus it cannot be strongly deformed to the top origin. -- This is very informal outline of a proof , I hope its good enough to help. -cwd1 —Preceding unsigned comment added by Cwd1 (talk • contribs) 17:35, 11 October 2007 (UTC)

I couldn't make sense of this example... sorry, maybe I am not used to working with this notion. An example I could came up with is the following: take the subset of the square IxI defined as: I x ( {0} U {1/n : n is a natural number} ) -- this is the union of the base horizontal segment and countably many vertical segments leaving from the base, one for each 1/n, plus the vertical segment corresponding to the left side of the square. Try to draw it, it is much easier to do so than to describe it! It is some sort of "infinite comb". There is a retraction to the top left corner {0,1} of the square: a point {1/n,y} in a tooth of the comb is firs sent down the tooth, then left until the side of the square and finally up on the side to {0,y}, to the same height it was before. This is clearly a deformation retract, but it can't be strong: the point {0,1} on the left side has to go down to {0,0} and then up again. —Preceding unsigned comment added by Alvisetrevi (talk • contribs) 08:48, 24 April 2008 (UTC)

I'd like to see some explicit examples of deformation retractions, if anyone can write one down. Druiffic (talk) 20:38, 24 January 2009 (UTC)

A space that deformation retracts to a point is contractible, however the converse is not true. For example, see Hatcher's book: Chapter 0, Exercise 6(b) (an explanation/solution can be found here. It is important to note that the dense subset of rational numbers inherits its subspace topology from $\mathbb {R}$). And when we say "deformation retracts to a point", there is no distinction between strong and weak deformation retracts. Hence, I am changing the second statement under the "properties" section to it's original (see this diff). - Subh83 (talk | contribs) 23:43, 30 November 2011 (UTC)

Of course the distinction between deformation retract and strong deformation retract should be made in the sentence you reference, because this article distinguishes the notions. You are citing Hatcher to support removal of the word "strongly", but as explicitly pointed out earlier in the article, his notion of deformation retract is a strong deformation retract in the sense of this article. The sentence without the word "strongly" would be fine if it were written inside of Hatcher's book, but it is not. It is written in the context of this article.128.196.226.219 (talk) 21:09, 25 January 2013 (UTC)

Retraction v/s Idempotency

How is a retraction different from an idempotent map on $X$? If $r:X\rightarrow A$ is a retraction then $r(r(x))=r(x)$ because $r(x)\in A$. OTOH, if $r$ is idempotent then for all $a$ in the codomain of $r$ there's some $x$ such that $r(x)=a$ therefore $r(a)=r(r(x))=r(x)=a$. The two definitions seem therefore equivalent, yet the respective articles don't even mention each other. —Preceding unsigned comment added by 192.114.107.4 (talk) 07:54, 16 September 2008 (UTC)

Well, what you say would be true only if your idempotent map was 1) surjective 2) continuous. Indeed, you're stating "for all $a$ in the codomain of $r$ there's some $x$ such that $r(x)=a$". But what if not all the elements of $A$ lie in the codomain of $r$?. A practical example: the absolute value from complex numbers to real numbers. It is idempotent, but definitely not a retraction (e.g. $r(-5)=5\neq -5$). Alvisetrevi (talk) 21:37, 10 October 2008 (UTC)

attributing sources

Since when do we give attributions and licensing information on a wikipedia page? Either the info we put is free, or if it isn't and needs attribution, it shouldn't be used.--24.85.90.31 (talk) 02:35, 30 November 2011 (UTC)