In his book: Bordism of diffeomorphisms and related topics [Lect. Notes Math. 1069 (1984; Zbl 0542.57001)], the second author computed the groups \(\Delta_ m\) of bordism classes of orientation preserving diffeomorphisms of closed oriented m-dimensional manifolds. His method worked also for other bordism types such as Spin bordism without, however, leading to complete results there. For instance, he obtained an exact sequence \[ 0\to K\to \Delta^{Spin}_{8k+2}\to \Omega^{Spin}_{8k+2}\oplus \Omega^{Spin}_{8k+3}\oplus W_ - ({\mathbb{Z}};{\mathbb{Z}})\to 0 \] where \(W_ -({\mathbb{Z}};{\mathbb{Z}})\) is the appropriate Witt group and where K is a subgroup of \({\mathbb{Z}}/16\). In the present paper, it is shown that this sequence splits and that \(K={\mathbb{Z}}/16\); the latter is done by considering the 2-torus \(T^ 2\) for \(k=0\) and forming products with \(P^ 2({\mathbb{H}})\) to get at higher values of k.
As an interesting application, it is shown that the Rohlin invariant of a (not necessarily null-bordant) Spin manifold of dimension \(8k+3\) is integral.