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Hilbert's 3rd Problem and Invariants of 3–manifolds

Walter D Neumann

Geometry & Topology Monographs 1 (1998) 383–411

DOI: 10.2140/gtm.1998.1.383

arXiv: math.GT/9712226


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 . 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.


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

Mathematical Subject Classification

Primary: 57M99

Secondary: 19E99, 19F27


Received: 21 August 1997
Published: 27 October 1998

Walter D Neumann
Department of Mathematics
The University of Melbourne
Vic 3052