Asian Journal of Mathematics

Volume 21 (2017)

Number 3

A new combinatorial class of $3$-manifold triangulations

Pages: 543 – 570



Feng Luo (Department of Mathematics, Rutgers University, New Brunswick, New Jersey, U.S.A.)

Stephan Tillmann (School of Mathematics and Statistics, University of Sydney, NSW, Australia)


We define a new combinatorial class of triangulations of closed $3$-manifolds, satisfying a weak version of $0$-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a $3$-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.


$3$-manifold, triangulation, $0$-efficient, circle-valued angle structure

2010 Mathematics Subject Classification

57M25, 57N10

Received 8 July 2014

Published 5 July 2017