Erratum: Homotopy self-equivalences of 4-manifolds (Q1024209)

Let \(M^4\) be a closed, oriented, smooth or topological \(4\)-manifold. The authors study the group \(\text{Aut}(M)\) of homotopy classes of homotopy self-equivalences of \(M\). Their main strategy is to compute \(\text{Aut}(M)\) by making use of various interlocking exact sequences. Let \(\mathcal{H}(M)\) denote the group of oriented \(h\)-cobordisms relative boundary from \(M\) to itself, and \(\widetilde{\mathcal{H}}(M)\) the group of such cobordisms equipped with a map \(F:W\to M\) which restricts to the identity on \(\partial_{-}W\) and a homotopy equivalence on \(\partial_{+}W\). Let \(B\) denote the \(2\)-type of \(M\). The main qualitative result of the paper is a commutative braid of exact sequences involving \(\text{Aut}(M)\), \(\text{Aut}(B)\), \(\widetilde{\mathcal{H}}(M)\), and certain spin cobordism groups of \(M\) and \(B\). In the case where \(\pi_1(M)\) is finite of odd order, this braid is computed explicitly. This yields a formula for \(\text{Aut}(M)\), given in terms of isometries of the quadratic \(2\)-type of \(M\) (see [\textit{I. Hambleton} and \textit{M. Kreck}, Math. Ann. 280, 85-104 (1988; Zbl 0616.57009)]) and the kernel of the second Stiefel-Whitney class of \(M\). The group \(\mathcal{H}(M)\) is determined as well for \(4\)-manifolds with odd order fundamental group, by means of a short exact sequence involving the isometries of the quadratic \(2\)-type of \(M\) and the structure group of smooth or topological manifold structures on \(M\times I\) relative to the given structure on \(\partial(M\times I)\).