We give an alternative proof of the Madsen–Weiss generalized Mumford conjecture. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem about folded maps. A technical ingredient in the proof is an –principle type statement, called the “wrinkling theorem” by the first and third authors [Invent. Math. 130 (1997) 345–369]. Let us stress the point that we are neither proving the wrinkling theorem nor the Harer stability theorem.
"Madsen–Weiss for geometrically minded topologists." Geom. Topol. 15 (1) 411 - 472, 2011. https://doi.org/10.2140/gt.2011.15.411