Manifolds Which Admit Maps with Finitely Many Critical Points Into Spheres of Small Dimensions

Abstract

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