# When can we say a homotopy restricted to a subset is a homotopy itself

1. Feb 26, 2013

### dumbQuestion

Say we have a homotopy H_t: X --> Y. Are there conditions that tell us when we can, say, restrict the homotopy to a subset A in X, and we will end up with a new homotopy?

I feel like I know how to go the other way around, like if (X,A) is a CW pair and I had a homotopy G_t: A --> Y, then I have the homotopy extension property, etc and can extend it to a homotopy from X --> Y. But what if I'm going the other way? I have a homotopy on the larger space, when can I restrict that homotopy to a subcomplex and still have a homotopy? Are there theorems regarding this?

I would be so appreciative of any insight

Last edited: Feb 26, 2013
2. Feb 26, 2013

### dumbQuestion

Re: When can we say a homotopy restricted to a subset is a homotopy it

I have found the following theorem on p.100 of a book called "introduction to homotopy theory" but i'm not sure it really answers the question because the notation is odd to me

th. 1.9: If f_0 homotopic to f_1 : K --> P (I'm assuming he's just saying, this is a homotopy F_t, where the first map is f_0 and the last map is f_1), and if the homotopy restrict to L , a subcomplex of K, is cellular, then the homotopy may be replaced by a cellular homotopy , extending the cellular homotopy on L.

Is this like saying, say we have a CW pair (X,A), and a homotopy F_t : X --> Y. Then say if we restrict our homotopy to A, and the image of all the resulting maps in the restricted homotopy are cellular (like say they all map to A, or to B which is a subcomplex of X), then we can actually replace F_t in the following way: 1) the restricted homotopy is now a homotopy H_t in its own right 2) use the HEP to extend this to F_t

Am I even on the right track with digesting this theorem?

3. Feb 27, 2013

### WWGD

Re: When can we say a homotopy restricted to a subset is a homotopy it

Well, I would say if your A is a segment of your cylinder Xx I , one could have a homotopy , but not one between the original functions and their graphs. You could then maybe get a homotopy from, f_k to f_k' , with 0<k<k'<1 (if your original ho0motopy is between f_0 and f_1).