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?