Erratum to “Stabilization for the automorphisms of free groups with boundaries”

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 G n these two isomorphic groups, the first diagram chase needs the composition Hi.Gn/!Hi.GnC1/!Hi.GnC1;G n/ to be trivial, which is the case because Gn and G n are conjugate in GnC1 . The second diagram chase needs the composition HiC1.Gn;Gn 1/!Hi.Gn 1/!Hi.G n/ 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!G n commuting with the inclusion of Gn 1 , so that the diagram chase is correct. For (A) and (B), there is an isomorphism Gn!G n that commutes with the inclusion of a subgroup H of Gn 1 , with H .H /ŠH .Gn 1/ in a range given by Theorem 2. Recall from [5] that M s n;k D N #.#nS S/#.#kS D/#.#sD/, where N is a fixed compact connected oriented 3-manifold, and that A n;k denotes the quotient of the mapping class group 0Diff.M s n;k rel @M s n;k / by twists along spheres embedded in M . Assertion (A) says that the map ̨i W Hi.A n;k /! Hi.A sC1 nC1;k /, induced by identifying discs in the last two boundary spheres of M , is surjective when n 3i and an isomorphism when n 3i C 2. Assertion (B) says that the same is true for the map ˇi W Hi.A sC2 n;k /!Hi.A s nC1;k /, where ˇi is induced by identifying the last two boundary spheres of M . Assertions (A) and (B) are used in [5] to show (1) and (3) in Theorem 4.1, namely that the stabilization maps i W Hi.Asn;k/!Hi.A s nC1;k /, induced by gluing