# The extending surfaces of immersions into surfaces

## TL;DR

This paper classifies all topological equivalence classes of immersed surfaces bounded by a given immersed circle in a closed oriented surface, extending previous results to more general cases with a new, straightforward method.

## Contribution

It introduces a new, simplified approach to classify immersed surfaces bounded by circles in closed surfaces, generalizing earlier work on immersed disks.

## Key findings

- Complete classification of immersed surfaces bounded by a circle in closed surfaces.
- A new method that simplifies the classification process.
- Extension of previous results to broader cases.

## Abstract

S. Blank solved the question of classifying immersed circles in $\mathbb{R}^{2}$ that extend to immersed disks, and how many topologically inequivalent disks can be extended. The quetions of various cases in $2$-dimension have already been solved by generalizing his method. In this paper, we give a new way, which is straightforward for the questions, and we determine all topological equivalence classes of immersed surfaces bounded by an arbitrary immersed circle in a closed oriented surface.