The dual boundary complex of the $SL_2$ character variety of a punctured sphere

TL;DR

This paper studies the topological structure of the $SL_2$ character variety of a punctured sphere, revealing that its dual boundary complex is homotopy equivalent to a sphere of a specific dimension.

Contribution

It establishes the homotopy type of the dual boundary complex for a broad class of $SL_2$ character varieties associated with punctured spheres.

Findings

Dual boundary complex is homotopy equivalent to a sphere of dimension 2(k-3)-1.

Provides a topological characterization of character varieties with generic conjugacy classes.

Advances understanding of the boundary structure in character varieties.

Abstract

Suppose $C_1,\ldots , C_k$ are generic conjugacy classes in $SL_2({\mathbb C})$. Consider the character variety of local systems on ${\mathbb P}^1-\{ y_1,\ldots , y_k\}$ whose monodromy transformations around the punctures $y_i$ are in the respective conjugacy classes $C_i$. We show that the dual boundary complex of this character variety is homotopy equivalent to a sphere of dimension $2(k-3)-1$.