Symmetries of 4-manifolds

This page collects notes and articles on symmetries of 4-manifolds.

Lecture notes

I gave a series of lectures in Warsaw in March 2024.
Here are lecture notes on Symmetries of 4-manifolds, by M. Powell.

Topological mapping class groups of simply-connected 4-manifolds.

Isotopy classes of diffeomorphisms of (k-1)-connected almost parallelizable 2k-manifolds by M. Kreck.
This paper classified diffeomorphism of simply-connected 4-manifolds up to pseudo-isotopy.
The topology of four-dimensional manifolds by M. Freedman.
Among many other things, this paper and the following book proved that every isometry of the intersection form of a simply-connected 4-manifold is realised by a self-homeomorphism.
Topology of 4-manifolds by M. Freedman and F. Quinn.
Ends of maps III by F. Quinn.
As well as establishing several foundational tools for studying 4-manifolds, this paper proved that the orientation-preserving mapping class group of the 4-sphere is trivial.
Pseudo-isotopies et isotopies en dimension quatre dans la categorie topologique by B. Perron.
This paper proved that pseudo-isotopy implies isotopy in the topological category for simply-connected 4-manifolds.
Pseudo-isotopies and isotopies in dimension four in the topological category by B. Perron.
Translation of Perron's article by Mark Powell.
Isotopy of 4-manifolds by F. Quinn.
This gave another proof that pseudo-isotopy implies isotopy in the topological category for simply-connected 4-manifolds, and the corresponding stable smooth fact.
Pseudo-isotopies of simply connected 4-manifolds by D. Gabai, D.T. Gay, D.H. Hartman, V. Krushkal, and M. Powell.
This gave a correction to one part of Quinn's proof from Isotopy of 4-manifolds.
Simply connected 4-manifolds with a given boundary by S. Boyer.
For simply-connected 4-manifolds with nonempty boundary, this described when a homeomorphism of the boundary extends to a homeomorphism of the entire manifold.
Simply connected 4-manifolds with a given boundary by R. Stong.
Stable mapping class groups of 4-manifolds with boundary by O. Saeki. This paper computed the smooth stable mapping class group for simply-connected smooth 4-manifolds with connected boundary.
Mapping class groups of simply connected 4-manifolds with boundary, by P. Orson and M. Powell.
Building on the papers above, this paper completed the calculation of the topological and stable-smooth mapping class groups of all compact, simply-connected 4-manifolds.
On the symmetries of the fake ℂP² by S. Kwasik.

Homotopy automorphisms of 4-manifolds.

On the homotopy theory of simply connected four manifolds by T.D. Cochran and N. Habegger.
Homotopy self-equivlences of 4-manifolds by I. Hambleton and M. Kreck.
An Erratum and an Addendum.

Mapping class groups of non-simply connected high-dimensional manifolds.

Pseudo-isotopies of compact manifolds by A. Hatcher and J. Wagoner.
What happens to Hatcher and Wagoner's formula when the first Postnikov invariant is nontrivial? by K. Igusa.
Pseudo-isotopies and the Bokstedt trace by B. Jahren.
Concordance spaces, higher simple homotopy theory, and applications by A. Hatcher.

Mapping class groups of non-simply connected 4-manifolds.

A note on homotopy and pseudoisotopy of diffeomorphisms of 4-manifolds by M. Krannich and A. Kupers.
Knotted 3-balls in S⁴ by R. Budney and D. Gabai.
On the automorphism groups of hyperbolic manifolds by R. Budney and D. Gabai.
Theta-graph and diffeomorphisms of some 4-manifolds by T. Watanabe.
The Casson-Sullivan invariant for homeomorphisms of 4-manifolds by D. Galvin.
Diffeomorphisms of 4-manifolds from graspers. by D. Kosanović.
Pseudo-isotopies and diffeomorphisms of 4-manifolds by O. Singh.
Smooth realisation of Hatcher-Wagoner obstructions in dimension four after stabilisation.
Grasper families of spheres in S² x D² and barbell diffeomorphisms of S² x S¹ x I by E. Fernández, D. Gay, D. Hartman, and D. Kosanović paper.
Reproves one of Singh's results by a different method.
On the Wh₁ obstruction and pseudo-isotopies of manifolds of dimension 3 and 4. by B. Jahren.
A preliminary report. There is an argument for realising the Z/2 invariants that has not been verified.
Second obstruction to pseudo-isotopy I by K. Igusa.
This is actually about the secondary Hatcher-Wagoner obstruction for a pseudo-isotopy to be isotopic to the identity.
On an argument of Igusa by R. Edwards.
Edwards' on Igusa's argument that a nontrivial mapping class of M x I rel. boundary gives rise to a nontrivial mapping class of M x S¹.
Second obstruction to pseudo-isotopy in dimension 3 by K. Igusa.
Low dimensional concordances, Whitney towers, and isotopies by S. Kwasik.
This paper gives an incomplete survey of Quinn's proof, and makes some unsubstantiated claims about realising Hatcher-Wagoner obstructions by pseudo-isotopies of 4-manifolds.
Mapping class groups of 4-manifolds with 1-handles by J. Lin, Y. Xie, and B. Zhang.
Dax invariant for closed embedded surfaces and the mapping class group of Σ x S² by J. Lin, Y. Xie, and B. Zhang.

Smooth mapping class groups of simply-connected 4-manifolds without gauge theory.

Symmetries of simply connected 4-manifolds, especially algebraic surfaces by S. Weintraub.
Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins by D. Gay.
Relations amongst twists along Montesinos twins in the 4-sphere by D. Gay and D. Hartman.
On Watanabe's theta graph diffeomorphism in the 4-sphere by D. Gay.
Corks for exotic diffeomorphisms by V. Krushkal, A. Mukherjee, M. Powell, and T. Warren.
On Torelli groups and Dehn twists of smooth 4-manifolds by M. Krannich and A. Kupers.
The 4-dimensional light bulb theorem by D. Gabai.
Pseudo-Isotopy and Diffeomorphisms of the 4-Sphere I: Loops of Spheres by D. Gabai, D. Gay, and D. Hartman.

Smooth mapping class groups of simply-connected 4-manifolds with gauge theory.

An obstruction to smooth isotopy in dimension 4 by D. Ruberman.
A polynomial invariant of diffeomorphisms of 4-manifolds by D. Ruberman.

Tautological classes of definite 4-manifolds by D. Baraglia.
On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds by D. Baraglia and H. Konno.
A gluing formula for families Seiberg-Witten invariants by D. Baraglia and H. Konno.
A note on the Nielsen realization problem for K3 surfaces by D. Baraglia and H. Konno.
Irreducible 4-manifolds can admit exotic diffeomorphisms by D. Baraglia and H. Konno.

The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary by H. Konno and M. Taniguchi.
From diffeomorphisms to exotic phenomena in small 4-manifolds by H. Konno, A. Mallick, and M. Taniguchi.
Diffeomorphisms of 4-manifolds with boundary and exotic embeddings by N. Iida, H. Konno, A. Mukherjee, and M. Taniguchi.
Non-smoothable homeomorphisms of 4-manifolds with boundary by D. Galvin and R. Ladu.
On localizing groups of exotic diffeomorphisms of 4-manifolds by H. Konno and A. Mallick. This paper should be contrasted with the results in "Corks for exotic diffeomorphisms".

The homology of moduli spaces of 4-manifolds may be infinitely generated by H. Konno.
This paper and the next appeared essentially simultaneously, and showed infinite generation of the mapping class groups of some simply-conencted 4-manifolds. They pointed out that Ruberman had only shown this for the Torelli group in his papers above.
On the mapping class groups of simply-connected smooth 4-manifolds by D. Baraglia.

Generalised Dehn twists.

The Dehn twist on a sum of two K3 surfaces by P. Kronheimer and T. Mrowka.
Isotopy of the Dehn twist on K3#K3 after a single stabilization by J. Lin.
Exotic Dehn twists on 4-manifolds by H. Konno, A. Mallick, and M. Taniguchi.
On four-dimensional Dehn twists and Milnor fibrations by H. Konno, J. Lin, A. Mukherjee, and J. Muñoz-Echániz.
The monodromy diffeomorphism of weighted singularities and Seiberg-Witten theory by H. Konno, J. Lin, A. Mukherjee, and J. Muñoz-Echániz.