Extended surgery theory for simply-connected $4k$-manifolds

Kreck proved that two $2q$-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q-1)$-smoothings over the same normal $(q-1)$-type (B,ξ). We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on Hq), when the manifolds are simply-connected, $q=2k$ is even and Hq(B) is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k-1)$-connected $4k$-manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected $4k$-manifold M with normal $(q-1)$-type (B,ξ) such that Hq(B) is free, an algebraic description of the stable class of M, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to M. Using this description, we explicitly compute the stable class of manifolds M with rank-$2$ hyperbolic intersection form.