Harer proved that the braid group Formula: see text is a duality group with a dualizing module called the Steinberg module StFormula: see text. He also gave a finite Formula: see text-module resolution of StFormula: see text that yields a finite free Formula: see text-module presentation of StFormula: see text, which we call the Harer presentation of StFormula: see text. This presentation has Formula: see text (the Formula: see textth Catalan number) generators and Formula: see text relations. In this article, we will simplify this presentation to arrive at a Formula: see text-module presentation of StFormula: see text with one generator and Formula: see text relations.
Broaddus et al. (Fri,) studied this question.