Celebratio Mathematica — Neumann

A celebration of Walter Neumann

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W. D. Neumann: “Hilbert’s 3rd problem and invariants of 3-manifolds,” pp. 383–411 in The Epstein birthday schrift. Edited by I. Rivin, C. Rourke, and C. Series. Geometry and Topology Monographs 1. Geometry and Topology Publishers (Coventry, UK), 1998. Dedicated to David Epstein on the occasion of his 60th birthday. MR 1668316 Zbl 0902.57013 ArXiv math/9712226 incollection

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 \( \mathbb{C} \). 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.

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Colin P. Rourke	Related	Volume
Caroline Mary Series	Related