# Problem 5 of Milnor's “Topology From The Differentiable Viewpoint”

I'am trying to come up with a solution to the referred problem which, by the way, states the following:

If $m<p$, show that every map $M^m\to S^p$ is homotopic to a constant map.

From the chapter, you may (must) assume $M$ is a compact (and, may be, boundaryless) smooth ($=C^\infty$) manifold (Hausdorff, second countable, locally euclidean topological manifold of dimension $m$). I've been trying to use Milnor's Theorem B:

Two mappings from $M$ to $S^p$ are smoothly homotopic if and only if the associated Pontryagin manifolds are framed cobordant.

I can't come up with a framed manifold corresponding to a constant function. Does it make any sense? I mean, what are the the regular values of such functions? What can you do with an empty set, in this case?

I've lots of questions here, if all I said above is completely nonsense, how to proceed then? In what other books could I learn more (I've googled a lot and all other references seemed to copy Milnor's work...)

Thanks in advance.

• Do you know Sard's theorem? – Tim kinsella Aug 8 '16 at 1:52
• It asserts that the set $f(M)$ has measure zero in $S^p$, right? Still don't get it, but already saw I can't apply Theorem B I was speaking above due to the inequality $m<p$. – user255306 Aug 8 '16 at 1:58
• Yes, thats how you show $f$ is not surjective in the first line of Tsemo's answer below. – Tim kinsella Aug 8 '16 at 2:14

## 1 Answer

If $m<p$, then the image of $f$ is not surjective, let $y\in S^p$ which is not in $f(M)$ and $p_y:S^p\rightarrow R^p$ the stereographic projection centred at $y$, $p_y\circ f$ is homotopic to a constant: define $H_t(x)=t(p_y\circ f)(x)$, $p_y^{-1}\circ H_t$ is the requested homotopy.

I use the fact that the stereographic projection $p_y:S^p-\{y\}\rightarrow R^p$ is a diffeomorphism

• All right, man! This puts an end to these question. Thanks, everybody, for the help! – user255306 Aug 8 '16 at 3:25