Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

TL;DR

This paper investigates smooth maps from certain high-dimensional manifolds into connected sums of $S^1\times S^n$, focusing on manifolds with finitely many critical points and determining their minimal number in specific dimensions.

Contribution

It provides explicit examples and computes the minimal number of critical points for smooth maps in dimensions (4,3), (8,5), and (16,9).

Findings

Identifies manifolds with smooth maps having finitely many critical points.

Calculates minimal critical points for dimensions (4,3), (8,5), and (16,9).

Contributes explicit examples to the theory of smooth maps with controlled critical points.

Abstract

We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\times S^n$ with only finitely many critical points, for $n\in\{2,4,8\}$, and compute the minimal number of critical points.