The disc embedding theorem

Here is a collection of material on the disc embedding theorem of Michael H. Freedman.

Original material

This book contains Three lectures on new infinite constructions in 4-dimensional manifolds by A. Casson. These are the lectures that defined Casson handles and showed them to have the right proper homotopy type.

The topology of four-dimensional manifolds by M. H. Freedman, Journal of Differential Geometry, 1982. The original Fields medal winning paper, in which it is proven that Casson towers are homeomorphic to standard handles.

The topology of 4-manifolds by M. H. Freedman and F. Quinn. The book. Chapters 1-5 prove the disc embedding theorem using gropes. A new version of this proof is given in the lecture notes below.

Four manifold theory, Durham conference proceedings, edited by C. Gordon and R. Kirby 1983. These proceedings contain many interesting papers connected to Freedman's advances. In particular, Ancel's write up of the sphere to sphere theorem and Gompf and Singh's refinement of reimbedding for Casson towers relate to the proof of the disc embedding theorem.

La conjecture de Poincaré topologique en dimension 4 by L. Siebenmann. This paper gives Siebenmann's account of Freedman's proof, minus Casson tower reimbedding.

The Poincaré conjecture in dimension 4 by L. Siebenmann, translated into English by Min Hoon Kim and M. Powell.

Siebenmann's paper together with the Gompf-Singh version of Casson tower reimbedding and the Ancel sphere to sphere write up, both from the Durham conference proceedings, provide another independent account of the disc embedding theorem.

Shrinkability of Bing-Whitehead decompositions by F. Ancel and M. Starbird, Topology 1989. This result is used in the grope proof of the disc embedding theorem.

Notes from 2013 onwards

Bing topology and Casson handles by S. Behrens, and many others. Notes from Freedmans 2013 lectures in Santa Barbara, roughly typed by the Bonn audience. The videos of the lectures are available here. The notes are currently undergoing extensive revision and a new version will be posted soon.

Shrinking of toroidal decomposition spaces by D. Kasprowski and M. Powell, Fundamenta Math. 2014. Generalised the Ancel-Starbird result to arbitrary links.

The class of good groups

The best class of groups for which surgery and s-cobordism are known is sub-exponential growth groups. The class of good groups is closed under extensions and colimits.

4-manifold topology I. Subexponential groups by M. Freedman and P. Teichner, Inventiones Math. 1995.

Subexponential groups in 4-manifold topology by V. Krushkal and F. Quinn, Geometry and Topology 2000.

Slicing links with the disc embedding theorem

A new technique for the link slice problem by M. Freedman, Inventiones Math. 1985.

Whitehead-3 is a slice link by M. Freedman, Inventiones Math. 1988.

Link composition and the topological slice problem by M. Freedman, Topology 1993.

4-manifold topology II. Dwyer's filtration and surgery kernels by M. Freedman and P. Teichner, Inventiones Math. 1995.

Casson towers and slice links by Jae Choon Cha and M. Powell, Inventiones Math. 2015.

Attempts to obstruct surgery for free groups

The main attempt to find obstructions has been the AB slice problem.

A geometric reformulation of 4-dimensional surgery by M. Freedman, Topology and its Applications 1986.

Are the Borromean rings A-B-slice by M. Freedman, Topology and its Applications 1986.

On the AB slice problem by M. Freedman and X. S. Lin, Topology 1988. Obstructions to generalised Borromean rings being AB slice with certain standard AB decompositions.

A counterexample to the strong version of Freedman's conjecture by V. Krushkal, Annals of Math. 2008. Without the standard hypothesis, there exists a solution to the AB slice problem.

For more papers on this topic, see Slava Krushkal's website.

Engel relations in 4-manifold topology by M. Freedman and V. Krushkal Forum of Math, Sigma, 2016.

A homotopy + solution to the AB slice problem by M. Freedman and V. Krushkal, Journal of Knot theory and its ramifications, 2016. The two previous papers show that the AB slice problem has a link homotopy solution.

Poincaré transversality and four-dimensional surgery by M. Freedman, Topology 1988. Another reformulation of the disc embedding conjecture.

The Round Handle Problem

The Round Handle Problem by Min Hoon Kim, M. Powell and P. Teichner. The RHP, proposed by Freedman and Krushkal in 'Engel relations in 4-manifold topology,' gives a new way to try to obstruct surgery for free groups. It cannot at present be solved by the Engel paper ideas.