The last time we proved that, for $g\geq 1$, there is a canonical isomorphism $$\text{MCG}^\pm (\Sigma_g)\to \text{Out}\big(\pi_1(\Sigma_g)\big),$$ under the assumption that closed orientable surfaces are topologically rigid, i.e., every homotopy equivalence between two closed orientable connected surfaces is homotopic to a homeomorphism.
Today we will start proving the topological rigidity of closed orientable surfaces.

A (short) sketch of the proof (for $g\geq 2$) is in the book "A primer on Mapping Class Groups" written by Benson Farb and Dan Margalit on page 237. We'll give the details of their steps and the other two cases, namely $g=0,1$, which we'll consider later.