$SKK^{\xi}_n$-groups of manifolds

Manifolds with a tangential structure $\xi: B \to BO$ can be cut and glued back twisted by a $\xi$-diffeomorphism along separating codimension one $\xi$-submanifolds. This gives rise to certain groups $SKK_n^{\xi}$ of $n$-dimensional $\xi$-manifolds, modulo cut-and-paste relations (Schneiden und Kleben). Kreck, Stolz, and Teichner have, under some mild assumptions, provided a description of the kernel of the surjection $SKK_n^\xi \to \Omega_n^\xi$ in an as-yet unpublished paper. We provide an alternative geometric proof of their result in a certain special case. We calculate the $SKK_n^\xi$ groups for certain tangential structures in terms of the corresponding bordism group by determining whether the corresponding short exact sequence splits. This is joint work with Renee Hoekzema and Luuk Stehouwer.