Manifolds which admit maps with finitely many critical points into spheres of small dimensions

We construct, for $m\geq 6$ and $2n\leq m$, closed manifolds $M^{m}$ with finite nonzero $\varphi(M^{m},S^{n}$), where $\varphi(M,N)$ denotes the minimum number of critical points of a smooth map $M\to N$. We also give some explicit families of examples for even $m\geq 6, n=3$, taking advantage of the Lie group structure on $S^3$. Moreover, there are infinitely many such examples with $\varphi(M^{m},S^{n})=1$. Eventually we compute the signature of the manifolds $M^{2n}$ occurring for even $n$.