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

TL;DR

This paper constructs specific high-dimensional manifolds with finite, nonzero minimum critical points for smooth maps into spheres, providing explicit examples and computing signatures for certain cases.

Contribution

It introduces new constructions of manifolds with finitely many critical points into spheres, including explicit families and cases with exactly one critical point.

Findings

Existence of manifolds with finite nonzero critical points for maps into spheres.

Explicit examples using Lie group structures on $S^3$.

Computed signatures for certain even-dimensional manifolds.

Abstract

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 $n$.