# Jonathan Hillman

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 Level 6, Room 609 Carslaw Building jonathan.hillman@sydney.edu.au +61 2 9351 5775 +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).

#### 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 29 June 2020; in particular Chapter 15 has been rewritten (yet again!). (See also the Errata and Addenda for the current (2020) revision, begun 31 July 2018).

#### 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

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.