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Let $A\subset B$ be paracompact spaces, and let $C$ be a paracompact space.

Let $f_0,f_1:A\rightarrow C$ be continuous functions. $F:A\times[0,1]\rightarrow C$ a homotopy from $f_0$ to $f_1$. Suppose $f_0,f_1$ can be extended continuously to functions $\tilde{f_0},\tilde{f_1}:B\rightarrow C$.

Under this hypothesis, can I find an extension of $F$ into $\tilde{F}:B\times[0,1]\rightarrow C$, homotopy from $\tilde{f_0}$ to $\tilde{f_1}$?

If false, can I recover the result by substituting $B$ with a neighbourhood $N$ of $A$?

Does it help to know there is already an homotopy $F':B\times [0,1] \rightarrow C$ with $F'_0|_A=F_0=f_0$ and $F'_1|_B=F_1=f_1$ ?

Are other hypothesis on $A,B,N,C$ necessary or is the affirmation totally false?

Motivation I have a homotopy $F$ defined on $A$ and a homotopy $F'$ defined on $B$ that are compatible (via restriction) at time $t=0,1$. I would like to find a third homotopy $F''$ defined on $B$ that restricts to $F$ at every time $t\in[0,1]$. I have a good idea on how to go doing this with partitions of unity but I need to be able to extend $F$ in a neighbourhood outside $A$.

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  • $\begingroup$ Probably you want the inclusion $A\subseteq B$ to be a cofibration. $\endgroup$
    – Mark Grant
    Commented Jul 27, 2015 at 8:33
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    $\begingroup$ This is possible if the the inclusion of A into B is a closed cofibration and a homotopy equivalence. $\endgroup$
    – Ricardo Andrade
    Commented Jul 30, 2015 at 5:31

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