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
Office: Room 617 Carslaw Building
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. (See Some questions on subgroups of 3-dimensional Poincaré duality groups for a problem list motivated by what is known for 3-manifold groups.)

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).


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.

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 19 April 2018; in particular Chapter 15 has been rewritten, to include new work completing the determination of virtually solvable 2-knot groups. (See also the Errata for the 2014 revision, begun 14 July 2014).

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".

Poincar\'e duality in low dimensions are the background notes for a four-lecture hour minicourse given in Agua de Lindoias, S.P., Brazil (29 July -- 3 August, 2012), and repeated (with a somewhat different presentation) in Chicago (April 2014).

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

$PD_4$-complexes and 2-dimensional duality groups is a synthesis of three of my papers on the notion of minimal models for $PD_4$-complexes.

See also Publications