Jonathan Hillman

Penrose tiling Honorary Reader in the School of Mathematics and Statistics at the University of Sydney.


Postal address: Dr Jonathan Hillman
School of Mathematics and Statistics F07
University of Sydney NSW 2006
Australia
Office: Level 6, Room 609 Carslaw Building
Email: jonathan.hillman@sydney.edu.au
Telephone: +61 2 9351 5775
Department Fax: +61 2 9351 4534

Research Interests

I am interested in applications of algebra to low dimensional topology, (2-complexes, 3- and 4-manifolds) and knots and links (in all dimensions). I am particularly interested in the interactions between the fundamental group and Poincaré duality. In particular, I believe that all 3-dimensional Poincaré duality groups are 3-manifold groups, although I cannot yet prove this.

Although I am now retired, and am no longer involved in supervising candidates for higher degrees, I remain a member of the Geometry and Topology group.

For one characterization of "Reader" (at an older university) see ``The Gaudy" by J.I.M.Stewart (page 218 in the Methuen paperback edition).

Books

Algebraic Invariants of Links (2nd edition, World Scientific Publishing Co, xiv+353pp, June 2012) is intended as an introduction to links and a reference for the invariants of abelian coverings of link exteriors, and to outline more recent work, particularly that related to free coverings, nilpotent quotients and concordance. The table of contents, Preface and Chapter 1 are available here as a .pdf file. (The second edition has two new chapters, on twisted polynomial invariants and on singularities of plane curves.) See also the Errata and Addenda for the first edition, and Errata and Addenda for the second edition.

Four-Manifolds, Geometries and Knots (Geometry and Topology Monographs, vol. 5, Geometry and Topology Publications, December 2002) is based on my 1989 and 1994 monographs on 2-knots and on geometric 4-manifolds. However the arguments have been improved in many cases, notably in using Bowditch's homological criterion for virtual surface groups to streamline the results on surface bundles, using $L^2$-methods instead of localization, completing the characterization of mapping tori, relaxing the hypotheses on subgroups of the fundamental group and in deriving the results on 2-knot groups from the work on 4-manifolds.

Revisions were made available through GT in 2007 and 2014. These incorporate new material, particularly in Chapters 4, 9, 10, 12, 16 and 18, and corrections to all the errors and typos found up to [30 June 2014]. See page xiv for a summary of the main changes. The version available here was last updated on 10 March 2021. (See also the Errata and Addenda for the current revision, begun 31 July 2018).

My most recent (final?) book is ``Poincaré Duality in Dimension Three", which has been published by MSP as an Open Book. The corrections noted to date (10 March 2021) may be found in Errata and Addenda.

Expository Material

Homology and Fundamental group are handouts for the Honours Course on Algebraic Topology.

Graphs, Surfaces and Knots corresponds to half of the third-year course ``Geometry and Topology".

Some questions on low dimensional topology is a list of problems in low dimensional topology, group theory and knot theory that I revisit regularly.

Other publications

Some corrections gives some corrections to various of my papers.

Aspherical 4-manifolds with elementary amenable fundamental groups grew out of discussions with Jim Davis at the MATRIX meeting on {\it Topology in Low and High Dimensions} at Creswick, Vic., in January 2019.

See also Publications