On the dimension of Harer's spine for the decorated Teichm\"uller space

TL;DR

This paper corrects the previously computed dimension of Harer's spine for the decorated Teichmüller space, clarifying the topological structure of these moduli spaces.

Contribution

It identifies and rectifies errors in Harer's original dimension calculation, providing the accurate dimension for the spine of the decorated Teichmüller space.

Findings

Corrected the dimension of Harer's spine for certain surfaces

Clarified the topological structure of the decorated Teichmüller space

Enhanced understanding of the moduli space topology

Abstract

In \cite{Ha86} Harer explicitly constructed a spine for the decorated Teichm\"uller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by $1$ and give the correct dimension.