Let $f:S^{n-1}\rightarrow Y$ be a continuous map from the sphere to a topological space. Why does $f$ have to be nullhomotopic for it to be extendable to the disk? I know this may be a silly question but I don't quite get it.
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If $f$ can be extended to the entire disk, then because the disk is contractible, there exists a homotopy from $f$ to the constant map as follows.
If $\tilde{f}\colon D^n\to Y$ is the extension of $f$ and $H\colon D^n\times I\to Y$ is a homotopy from $\tilde{f}$ to the constant map $c\colon D^n\to Y$ (which exists because $D^n$ is contractible), then we define $H'\colon S^{n-1}\times I\to Y$ by $H'(x,t)=H(x,t)$ which is a homotopy by construction from $f$ to $c|_{S^{n-1}}$.
Dan Rust
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Thank you! I was overlooking the fact that the disk is contractible. – Sak Jul 01 '14 at 17:01