# Smooth maps with singularities of bounded K-codimensions

## TL;DR

This paper establishes the relative homotopy principle for smooth maps with controlled singularities and explores the structure of homotopy self-equivalences of manifolds based on singularity codimensions.

## Contribution

It proves the relative homotopy principle for maps with specific singularities and analyzes the filtration of self-equivalence groups by singularity codimensions.

## Key findings

- Proved the relative homotopy principle under mild conditions.
- Analyzed the filtration of the group of homotopy self-equivalences.
- Connected singularity types with manifold self-equivalence structures.

## Abstract

We will prove the relative homotopy principle for smooth maps with singularities of a given {\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {\cal K}-codimensions.