Extension of maps on $\partial D^n $ to maps on $D^n$

Question

I'm trying to understand the extension of maps on $\partial D^n $ to maps on $D^n$

1. For a map $f:\partial D^n\mapsto X$, when can we say this map extends to a map $f': D^n \mapsto X$?

2. For two maps $f_1,f_2: D^n\mapsto X$, if they induce the same map on $\partial D^n$, when can we say that they are thus homotopic maps?

Also, does the above things actually comes from some larger setting rather than just for pair $(\partial D^n, D^n)$? Any help is appreciated.

Answer 1

1. By definition, a map $f:\partial D^n\to X$ extends to a map $f':D^n\to X$ if and only if $f$ is null-homotopic, i.e. homotopic to a constant map. See, for example, this question.

2. This is false in general (see this question, for example). However, in some special cases, we have results such as Alexander's Trick, which may be of interest.