Is every map $N \to \mathbb{S}^m$ null-homotopic if $N$ is a compact $n$-dimensional manifold and $n < m$?

Question

We know that $\pi_n(\mathbb{S}^m) \cong 0$ for $n < m$, i.e every map from $\mathbb{S}^n \to \mathbb{S}^m$ is null-homotopic. From this can we conclude that every map from $N \to \mathbb{S}^m$ is null-homotopic if $N$ is a compact $n$-dimensional manifold?

Answer 1

Every compact manifold is homotopy equivalent to a CW complex, so your statement follows from the theorem of cellular approximation.