Geometry & Topology Monographs 1 (1998), The Epstein Birthday Schrift, paper no. 19, pages 383-411.

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

Walter D Neumann

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

AMS subject classification. Primary: 57M99. Secondary: 19E99, 19F27.

E-print: arXiv:math.GT/9712226

Submitted: 21 August 1997. Published: 27 October 1998.

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Walter D Neumann
Department of Mathematics, The University of Melbourne
Parkville, Vic 3052, Australia

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