I came across this post: Homotopy groups of compact topological manifold which states exactly the result I need for a theorem I'm working on. However, I would need a reference, since the audience need not be very well-versed in homotopy theory.
Could someone suggest where I can find the result:
Theorem: Every closed, connected smooth $d$-manifold $M$ has a continuous and not nullhomotopic map $f: S^{d'} \rightarrow M$ for some sphere $S^{d'}$ with $1 \leq d' \leq \dim(M)$.
In other words, if $M$ is a closed and connected smooth manifold then there is a non-trivial $\pi_{d'}(M)$ for some $d'\leq \dim(M)$.
This is not a reference but a short proof:
if not, then with $d'=1$ we see that $M$ would have to be simply-connected.
In particular, if its homology groups all vanish, then $M$ is contractible. But the homology groups in dimension $> \dim(M)$ always vanish, and the hypothesis implies (by Hurewicz) that the homology groups in dimension $\leq \dim(M)$ vanish too.
This implies that $M$ is contractible, which is impossible by Poincaré duality (either mod $2$, or integrally because $M$ is simply-connected)
More simply put: $M$ is mod $2$-orientable, so it must have nontrivial mod $2$-cohomology, this must be in dimension $\leq \dim(M)$, but the hypothesis implies it doesn't, by the Hurewciz theorem.