# Singular Patterns of Generic Maps of Surfaces with Boundary into the Plane

## TL;DR

This paper presents an explicit algorithm for simplifying generic maps of surfaces with boundary into the plane by minimizing cusps and singular components, with applications in topological field theory.

## Contribution

It introduces a local modification algorithm that reduces cusps and computes minimal singular components for maps with fixed boundary conditions and prescribed singular patterns.

## Key findings

- Number of cusps is invariant mod 2 and can be reduced to at most one.
- Minimal number of singular components is explicitly computed.
- Algorithm applies to pseudo-immersions and topological field theory computations.

## Abstract

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the combinatorial information of how the 1-dimensional singular locus meets the boundary. Each step of our algorithm modifies the given map only locally by either creating or eliminating a pair of cusps. We show that the number of cusps is an invariant modulo 2 and can be reduced to at most one, and we compute the minimal number of components of the singular locus in terms of the prescribed data. Applications include a discussion of pseudo-immersions as well as the computation of state sums in Banagl's positive topological field theory.