# A bijection between $m$-cluster-tilting objects and $(m+2)$-angulations in $m$-cluster categories

## TL;DR

This paper establishes a bijection between geometric $(m+2)$-angulations and $m$-cluster tilting objects in certain categories, linking flips to mutations and using reductions to extend results.

## Contribution

It provides a new geometric interpretation of $m$-cluster categories of Dynkin types, connecting angulations with tilting objects and mutations.

## Key findings

- Bijection between $(m+2)$-angulations and $m$-cluster tilting objects
- Flips in angulations correspond to mutations of tilting objects
- Reduction techniques relate small cases to general categories

## Abstract

In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic $m$-cluster tilting objects. Under these bijections, flips of $(m+2)$-angulations correspond to mutations of $m$-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the $m$-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows to infer results from small cases to the general ones.