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 $(B,\xi)$. 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 $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ 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 $(B,\xi)$ such that $H_q(B)$ 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.