Extended surgery theory for simply-connected 4 k -manifolds

Kreck proved that two 2 q -manifolds are stably diffeomorphic if and only if they admit normally bordant normal left parenthesis q minus 1 right parenthesis ( q − 1 ) (q-1) -smoothings over the same normal left parenthesis q minus 1 right parenthesis ( q − 1 ) (q-1) -type left parenthesis upper B comma xi right parenthesis ( B , ξ ) (B,ξ) . We show that ‘stably diffeomorphic’ can be replaced by ‘diffeomorphic’ if the normal smoothings have isomorphic Q-forms (consisting of the intersection form of the manifold and the induced homomorphism on upper H Subscript q H q Hq ), when the manifolds are simply-connected, q equals 2 k q = 2 k $q=2k$ is even and upper H Subscript q Baseline left parenthesis upper B right parenthesis H q ( B ) 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 left parenthesis 2 k minus 1 right parenthesis ( 2 k − 1 ) (2k-1) -connected 4 k -manifold with the kernel of a certain bordism map. By the calculations of Senger and Zhang and earlier results, these ke