Geometry & Topology Monographs 1 (1998) 383–411

DOI: 10.2140/gtm.1998.1.383

arXiv: math.GT/9712226

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.

Keywords

scissors congruence, hyperbolic manifold, Bloch group, dilogarithm, Dehn invariant, Chern–Simons

Mathematical Subject Classification

Primary: 57M99

Secondary: 19E99, 19F27

References

Publication

Received: 21 August 1997
Published: 27 October 1998

Authors