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?(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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