If $f,g$ are two maps from $(D^n,S^{n-1})$ to $(D^n,S^{n-1})$ such that they have the same degree, that is $f_*[\mu]=g_*[\mu]$ where $[\mu]$ is a generator of $H_n(D^n,S^{n-1})$, then can we find a homotopy $f_t:(D^n,S^{n-1}) \rightarrow(D^n,S^{n-1})$ between $f$ and $g$?
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by the long exact sequence for homotopy groups, the morphism $\pi_n(D^n,S^{n-1})\to\pi_{n-1}(S^{n-1})\cong\mathbb Z$ is an isomorphism. So the answer is yes. – user8268 Oct 12 '13 at 20:33
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When $n>2$, we can use Hurewicz theorem directly. When $n=1$, the conclusion is true since it is orientation-preserving and we can set $f_t=(1-t)f+tg$. But I am not sure about the case when $n=2$, since $\pi_1(S^1)=\mathbb Z$ which can act on $\pi_2(D^2,S^1)$. – Summer Oct 12 '13 at 23:18