Celebratio Mathematica

W. D. Neumann and D. Zagier: “Volumes of hyperbolic three-manifolds,” Topology 24 : 3 (1985), pp. 307–332. MR 815482 Zbl 0589.57015 article

By “hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold \( M \) of finite volume. By Mostow rigidity the volume of \( M \) is a topological invariant, indeed a homotopy invariant, of the manifold \( M \). There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [Borel 1981; Zagier 1986]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [1977]. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes.

W. D. Neumann: “Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3-manifolds,” pp. 243–271 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184415 Zbl 0768.57006 incollection

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

@incollection {key1184415m,
	AUTHOR = {Neumann, Walter D.},
	TITLE = {Combinatorics of triangulations and
		 the {C}hern--{S}imons invariant for
		 hyperbolic 3-manifolds},
	BOOKTITLE = {Topology '90: {P}apers from the research
		 semester in low-dimensional topology
		 held at {O}hio {S}tate {U}niversity},
	EDITOR = {Apanasov, Boris and Neumann, Walter
		 D. and Reid, Alan W. and Siebenmann,
		 Laurent},
	SERIES = {Ohio State University Mathematics Research
		 Institute Publications},
	NUMBER = {1},
	PUBLISHER = {de Gruyter},
	ADDRESS = {Berlin},
	YEAR = {1992},
	PAGES = {243--271},
	URL = {https://www.degruyter.com/document/doi/10.1515/9783110857726/pdf#page=255},
	NOTE = {(Columbus, OH, February--June 1990).
		 MR:1184415. Zbl:0768.57006.},
	ISSN = {0942-0363},
	ISBN = {9783110125986},
}

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.

David Bernard Alper Epstein	Related
Colin P. Rourke	Related	Volume
Caroline Mary Series	Related

W. D. Neumann: “Extended Bloch group and the Cheeger–Chern–Simons class,” Geom. Topol. 8 : 1 (2004), pp. 413–474. MR 2033484 Zbl 1053.57010 ArXiv math/0307092 article

year	title	people

			clear

Walter D. Neumann

Hyperbolic 3-manifolds, the Bloch group,
and the work of Walter Neumann

Filter the Bibliography List

Hyperbolic 3-manifolds, the Bloch group, and the work of Walter Neumann

Filter the Bibliography List

Hyperbolic 3-manifolds, the Bloch group,
and the work of Walter Neumann