On the homotopy theory of simply connected four manifolds (Q749916)

Let X be a simply connected closed 4-manifold. The paper is concerned with the computation of HE(X), the group of homotopy classes of self- homotopy-equivalences of X, resp. \(HE^+(X)\), the homotopy self- equivalences which preserve orientation. Let \(Aut(H_ 2(X),\cdot)\) resp. \(Aut(H_ 2(X),\pm \cdot)\) be the group of automorphisms of the homology group \(H_ 2(X)\) which preserve (resp. preserve up to sign) the intersection form. Moreover, regard the 2nd Stiefel-Whitney class \(w_ 2(X)\in H^ 2(X;{\mathbb{Z}}_ 2)\) as a homomorphism \(H_ 2(X,{\mathbb{Z}}_ 2)\to {\mathbb{Z}}_ 2\). The group \(Aut(H_ 2(X),\pm \cdot)\) acts in an obvious way on \(\ker (w_ 2)\). Theorem: The group HE(X) resp. \(He^+(X)\) is isomorphic to the semi-direct product of \(Aut(H_ 2(X),\pm \cdot)\) resp. \(Aut(H_ 2(X),\cdot)\) and \(\ker (w_ 2)\). The proof of the theorem is purely homotopy-theoretic. Along the way, the authors give an explicit computation of \(\pi_ 4(X)\). They point out that their paper corrects several errors in the literature on 4- manifolds.