Stabilization for the automorphisms of free groups with boundaries, Geom. Topol

We correct the proof of Theorem 4.1 in our paper “Stabilization for the automorphisms of free groups with boundaries”. 20F28; 57M07 The purpose of this note is to point out a gap in an argument in our paper [5] and explain how to fill it (or bypass it). The gap in [5] occurs in Section 4 in the proof of assertions (A) and (B), at the point where there are diagram chasing arguments in two commutative diagrams (displayed on page 1333 for case (A)). In each diagram the groups Gn in the two rows are isomorphic but not identical. If we denote by Gn and G0n these two isomorphic groups, the first diagram chase needs the composition Hi.Gn/!Hi.GnC1/!Hi.GnC1;G0n / to be trivial, which is the case because Gn and G0n are conjugate in GnC1. The second diagram chase needs the composition HiC1.Gn;Gn1/!Hi.Gn1/!Hi.G0n / to be trivial, but there is no a priori reason for this to be true, although it is true and follows a posteriori from Theorem 2 below. An analogous diagram chase is used in the proof of assertion (C), but in that case there is an isomorphism Gn!G0n commuting with the inclusion of Gn1, so that the diagram chase is correct. For (A) and (B), there is an isomorphism Gn!G0n that commutes with the inclusion of a subgroup H of Gn1, with H.H /ŠH.Gn1 / in a range given by Theorem 2. Recall from [5] that M s n;k D N #.#nS1S2/#.#kS1D2/#.#sD3/, where N is a fixed compact connected oriented 3-manifold, and that As n;k denotes the quotient of the mapping class group 0Diff.M sn;k re