Abstract

The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. We can actually lift this to a scissors congruence spectrum which admits a map to the K-theory of Z, which on pi_0 recovers the Euler characteristic map. I will discuss what this higher homotopical lift of the Euler characteristic sees on the level of pi_1, and some speculative connections with the cobordism category. This is joint work in part with Hoekzema, Murray, Rovi and Semikina, and in part with Raptis and Semikina.