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

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In summary, the theorem states that if a homotopy between two functions f_0 and f_1 is cellular, then one can replace it with a cellular homotopy between their graphs.
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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
 
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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

http://books.google.com/books?id=hD...AEwCQ#v=onepage&q=homotopy restricted&f=false

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?
 
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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).
 

1. What is a homotopy?

A homotopy is a continuous transformation between two functions or maps. It is a type of mathematical concept used in topology to study the shape and properties of spaces.

2. What does it mean for a homotopy to be restricted to a subset?

When we say a homotopy is restricted to a subset, it means that the transformation is only occurring within a specific part of the space or function, rather than the entire space or function.

3. How do we determine if a homotopy restricted to a subset is still considered a homotopy?

In order for a homotopy to be considered valid when restricted to a subset, it must still satisfy the properties of a homotopy. This means that the transformation must be continuous and must maintain the same start and end points.

4. Why is it important to understand when a homotopy restricted to a subset is still considered a homotopy?

Understanding when a homotopy restricted to a subset is still considered a homotopy is important because it allows us to analyze and study the properties of the subset separately from the larger space. It also helps us to better understand the relationship between the subset and the larger space.

5. Can a homotopy restricted to a subset be considered a homotopy if the subset is disconnected?

In order for a homotopy to be considered valid, the subset must be connected. If the subset is disconnected, it is not possible for a continuous transformation to occur between the two functions or maps. However, it is possible for a homotopy to be restricted to a disconnected subset if the transformation occurs separately within each connected component.

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