# Maps with finitely many critical points into high dimensional manifolds

## TL;DR

This paper investigates smooth maps with finitely many cone-like singular points between closed manifolds, establishing conditions under which the domain manifold admits a fibration over the target and characterizing the minimal number of critical points.

## Contribution

It provides new topological conditions for the existence of fibrations and minimal critical points in maps between high-dimensional manifolds with finitely many singularities.

## Key findings

- Most such maps induce a locally trivial fibration.
- Except for specific low-dimensional cases, the domain admits a fibration over the target.
- There exists a smooth map with at most one critical point in the general case.

## Abstract

Assume that there exists a smooth map between two closed manifolds $M^m\to N^k$ with only finitely many cone-like singular points, where $2\leq k\leq m\leq 2k-1$. If $(m,k)\not\in\{(2,2), (4,3), (5,3), (8,5), (16,9)\}$, then $M^m$ admits a locally trivial topological fibration over $N^k$ and there exists a smooth map $M^m\to N^k$ with at most one critical point.