# Curves on surfaces and surgeries

## TL;DR

This paper investigates collections of curves on closed surfaces that cut the surface into a disk, introducing a surgery operation that connects any two such collections through a sequence of transformations.

## Contribution

It defines a surgery operation on curve collections and proves that any two collections can be connected via this operation, advancing understanding of surface topology.

## Key findings

- Defined a surgery operation on curve collections.
- Proved connectivity of all such collections via surgeries.
- Established a foundational result in surface topology.

## Abstract

We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries.