Journal article
Circle-valued angle structures and obstruction theory
CD Hodgson, AJ Kricker, RM Siejakowski
Journal of Topology and Analysis | WORLD SCIENTIFIC PUBL CO PTE LTD | Published : 2026
Abstract
We study spaces of circle-valued angle structures, introduced by Feng Luo, on ideal triangulations of 3-manifolds. We prove that the connected components of these spaces are enumerated by certain cohomology groups of the 3-manifold with ℤ2-coefficients. Our main theorem shows that this establishes a geometrically natural bijection between the connected components of the spaces of circle-valued angle structures and the obstruction classes to lifting boundary-parabolic PSL(2, ℂ)-representations of the fundamental group of the 3-manifold to boundary-unipotent representations into SL(2, ℂ). In particular, these connected components have topological and algebraic significance independent of the i..
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Awarded by Nanyang Technological University
Funding Acknowledgements
The computations of the connected components of the spaces of circle-valued angle structures were performed by the student Tracey Chen during a summer research Project at the University of Melbournein January-February 2020 under the supervision of Craig Hodgson. We thank her for her contribution. This research has been partially supported by Grants DP160104502 and DP190102363 from the Australian Research Council. The Project also received support through the AcRF Tier 1 Grants RG 32/17 and RG 17/23 from the Singapore Ministry of Education. Rafa lSiejakowski was supported by Grant #2018/12483-0 of the Sao Paulo Research Foundation (FAPESP). Our work has benefited from the support of Nanyang Technological University and the University of Melbourne. We would like to thank these institutions for their hospitality.