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Is there a transitive verb, in common use, which means 'deform via a homotopy'? I used to think 'homotope' was the answer, but it produces surprisingly few relevant matches on Google, so now I have my doubts. I want to be able to say things like "By Lemma 17 we can homotope the k-cells (rel boundary) into the subspace Y". I am well aware that I could rearrange the sentence to say something like "...there is a homotopy such that...", but that's not what I want to do. I want a simple, one-word verb with this meaning. I suppose 'deform' might work, but it's not as specific as I would like.

Another form of this question: Does the use of 'homotope' in the sample sentence of the previous paragraph sound strange? Sound standard and idiomatic?

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I just did a quick "grep -r homotope *" search in my TeX directory. Three of my papers use the word homotope in the way you use it. Sounds fine to me. I also say it quite a bit in talks. I use the word "isotope" vs "isotopy" in the same way, as well. That sounds a little more strange since isotope has another meaning. – Ryan Budney Apr 5 '10 at 3:53
If you can "google", then you can "homotope". The meaning is evident, so it's a word--linguistic prescriptivists be damned! – Sammy Black Apr 5 '10 at 4:05
It occurs 19 times in the bible, oops, I mean in my algebraic topology book (ha!). But the style of the book is a little informal. Of the 19, 12 are in the form "homotoped" as in "can be homotoped" or "has been homotoped". So obviously I think it's a fine word, and I wouldn't want to do without it. There's also the issue of how to pronounce it, the main choice being between a long or short final "o". I lean toward the long "o", which is why I spell it homotope rather than homotop, but this may be a British vs. American English thing. – Allen Hatcher Apr 5 '10 at 10:36
Zoran, the cat has been out of the box for some time in that this is a phrase I've been using since my undergrad years. I'm pretty sure I've heard senior figures like Siebenmann use this phrase. So this is likely as old a phrase as any in topology. Moreover, the context for which kind of homotopy is usually clear when the phrase is used so it's not confusing unless the author chooses to make it so. – Ryan Budney Apr 5 '10 at 18:31
We're fast discovering who's never read Allen's book! :) – HJRW Apr 5 '10 at 20:43

Since all of the responses thus far have been in the form of comments rather than answers, I'll summarize the results in this answer.

Of the first seven people to comment, all thought that "homotope" was a bit informal. Four commenters thought it was fine to use in print, while three thought it should be avoided in formal papers.

I did a full-text arXiv search, and found 408 occurrences of "homotoped" and 362 occurrences of "homotope". (I looked at only the mathematics sections, not physics etc.)

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i say go for it! – Sean Tilson Apr 6 '10 at 0:29

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