Hilbert's 3rd Problem and invariants of 3-manifolds

Abstract

This paper is an expansion of my lecture for David Epstein's birthday, which traced a logical progression from ideas of Euclid on subdividing polygons to some recent research on invariants of hyperbolic 3-manifolds. This `logical progression' makes a good story but distorts history a bit: the ultimate aims of the characters in the story were often far from 3-manifold theory. We start in section 1 with an exposition of the current state of Hilbert's 3rd problem on scissors congruence for dimension 3. In section 2 we explain the relevance to 3-manifold theory and use this to motivate the Bloch group via a refined `orientation sensitive' version of scissors congruence. This is not the historical motivation for it, which was to study algebraic K-theory of C. Some analogies involved in this `orientation sensitive' scissors congruence are not perfect and motivate a further refinement in section 4. Section 5 ties together various threads and discusses some questions and conjectures.