1997/12/31 by Walter D. Neumann
Mathematics · #math.GT #msc:57M99 #msc:19E99 #msc:19F27
published as Geom. Topol. Monogr. 1 (1998), 383-411 · 29 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper19.abs.html
arxiv created 1998/10/27 · arxiv updated 2009/11/30
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.