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.
Notes on file formats
Walter D Neumann
Department of Mathematics, The University of Melbourne
Parkville, Vic 3052, Australia
Email: neumann@maths.mu.oz.au
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