A Milnor Differential Topology Excercise

Question

If $m<p$, show that every map $f:M^m\longrightarrow\ S^p$ is homotopic to a constant, where $M^m$ is smooth manifold of dimension $m$.

I tried to show that $M^m$ is contractible or convex, but I couldn't. Any Idea would be helpful.

Answer 1

Expanding on John's hint.

If $f$ is not onto then there is a $x_0\in S^n\setminus f(M)$, so $f$ is actually a map from $M$ into $S^n\setminus\{x_0\}$. Using that $S^n\setminus\{x_0\}\equiv \mathbb R^n$ (here we are using stereographic projection) then $f$ is essentially a map from $M$ into $\mathbb R^n$ since $\mathbb R^n$ is contractible $f$ is homotopic to a constant.