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.
