Cut-and-paste invariants of manifolds via algebraic K-theory

Recent work of Zakharevich and Campbell has focused on developing the K-theoretic machinery to study scissors congruence problems and applying these tools to the Grothendieck ring of varieties.
In this talk we will discuss a new application of their framework to study the so called cut-and-paste invariants of manifolds. Namely, we will describe a K-theory spectrum, which recovers the classical groups SK_n ("schneiden und kleben" is German for "cut and paste") as its zeroth homotopy group.  We will also describe how the Euler characteristic, which is an example of a cut-and-paste invariant, fits into this new setup. We will also discuss what is conjecturally expected from the higher homotopy groups of the constructed spectrum and its potential connection to the classical notions such as Madsen-Tillmann spectra.

Joint work with R. Hoekzema, M. Merling, C. Rovi, L. Wells.