Scissors congruence K-theory of manifolds and cobordisms

The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in $\mathbb{R}^n$, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut-and-paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste").

In this talk I will introduce a parametrized version of scissors congruence K-theory of manifolds with tangential structure and discuss the relation of this K-theory spectrum with SK-invariants, the cobordism category and algebraic K-theory of spaces.