Lifting generic maps to embeddings. Triangulation and smoothing

TL;DR

This paper establishes conditions under which certain PL and smooth maps can be lifted to embeddings in higher-dimensional spaces, ensuring embeddings can be achieved through triangulation and smoothing techniques.

Contribution

It proves that lifts to topological embeddings imply lifts to PL or smooth embeddings under specific conditions, bridging topological and geometric embedding theories.

Findings

PL maps lifting to topological embeddings also lift to PL embeddings.

Stable smooth maps lifting to topological embeddings lift to smooth embeddings.

Results connect topological, PL, and smooth embedding theories.

Abstract

We show that if a non-degenerate PL map $f:N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there. We also show that if a stable smooth map $N^n\to M^m$, $m\ge n$, lifts to a topological embedding in $M\times\mathbb R$, then it lifts to a smooth embedding in there.