I will discuss the classification of so-called controlled cut-and-paste invariants for closed manifolds equipped with a tangential structure (oriented, unoriented, spin, etc.). These invariants are closely related to three more familiar objects: bordism groups, fundamental groups of the bordism categories and topological quantum field theories (TQFTs), as I will explain.
After introducing cut-and-paste invariants, I will describe how they relate to the homotopy type of the Madsen–Tillmann spectrum (a certain Thom space). Using this, I will show how Genauer's long exact sequence allows one to compute controlled cut and paste invariants. For oriented and unoriented structures, the resulting cut and paste groups recover classical computations from the 1970s. I will present new computations which apply for example in the spin case, which exhibit an 8-periodic phenomenon. A new invariant in this case is given by the mod 2 Kervaire semicharacteristic. Finally, I will explain how these invariants correspond precisely to (non-unitary) invertible TQFTs. No prior familiarity with TQFTs will be assumed. This is based on joint work with Renee Hoekzema and Simona Veselá.