Exotic 4-manifolds learning seminar

Autumn 2020

the seminar time and place schedule references contact

The seminar

This is a learning seminar about exotic 4-manifolds, aimed at graduate students. If you are interested do not hesitate to contact me.

Time and place

We meet online weekly on Mondays 17:00-19:00 London time (equivalently 12:00-14:00 New York time).
Every Monday a link is sent to the partecipants to join the seminar. The software we use is BigBlueButton which is completely free and open source.
The server that hosts us is freemath, mantained by Yankı.

Schedule of talks

Date	Speaker	Topic
October 1	Anubhav Mukherjee	Introduction to the seminar
October 5	Bruno R.S.Roso	Seiberg-Witten invariant
October 12	Bruno R.S.Roso	Seiberg-Witten applications
October 19	Fraser Binns	Operations/Constructions on 4-manifolds
October 26	Shengzhen Ning	Stein manifolds part 1
November 2	Anubhav Mukherjee	Stein manifolds part 2
November 9	Benjamin Ruppik	Some of Freedman's results on topological 4-manifolds
November 16	Diego Santoro & Ludovico Battista	Constructing small exotic manifolds
November 23	Roberto Ladu	Cork decomposition theorem
November 30	Stefan Mihajlovic	Corks part 2
December 7	Jie Min	Fintushel and Stern reverse engineering
December 14	Shuo Zhang	Akhmedov and Park construction

References

The main reference for this seminar is:
[A]Selman Akbulut. 4-Manifolds. Oxford University Press. 2016.

Stein manifolds
The generic reference is chapter 8. of [A]. Another useful reference may be Gompf-Stipsicz book (the last chapter is on Stein manifolds).
- Eliashberg characterization of cpt Stein 4-manifolds in terms of 1,2 handles
  Thm 8.11 in [A] pg. 100. (not proved in the book though) Main reference: Y.Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Int. J. Math. 1(1) (1990) 29-46.
- Convex decomposition.
  Thm 8.13 in [A] (proved in the book) the reference paper is S. Akbulut and R. Matveyev, A convex decomposition theorem for 4- manifolds, Int. Math. Res. Notices, no. 7 (1998) 371-381.
- Lisca-Matic (embedding into minimal Kahler)
  Thm 8.19 in [A] (only statement) the reference is P. Lisca and G. Matic, Tight contact structures and Seiberg-Witten invariants, Invent. Math.129 (1997) 509-525.
- Stein = |PALF| and compactification of a cpt Stein manifold to closed symplectic 1-connected manifold
  Thm 8.17 and 8.20 of [A] (fully proved in his book, notice it also requires some results of the previous chapter on Lefschetz fibrations). The references there cited are S. Akbulut and B. Ozbagci, Lefschetz fibrations on compact stein surfaces, Geom. Topol. 5 (2001) 319-334. S. Akbulut and B. Ozbagci, On the topology of compact Stein surfaces, Int. Math. Res. Notices 15 (2002) 769-782.
Seiberg-Witten
- Non vanishing theorem
  C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809-822.
- Genus bound
  P. Kronheimer and T. Mrowka, On genus of imbedded surfaces in the projective plane, Math. Res. Lett.1 (1994) 797-808.
- Self intersection of spheres in minimal Kahler
  R. Fintushel and R.J. Stern, Immersed spheres in 4-manifolds and the immersed Thom Conjecture, Turk. J. Math.19 (1995) 145-157, also in Proc. GGT 3 (1994) 27-39.
Topological 4-manifolds
Key results to us:
1. extension of a diffeo of boundary(X) to homeomorphism of X (under suitable hypothesis). This can be proven using a relative version of the TOP-h-cobordism theorem.
2. Alexander polynomial = 1 => TOP slice (uses the disk embedding theorem).
  M. Freedman, The topology of 4-dimensional manifolds, J. Differ. Geom. 17 (1982) 357-453.
  Michael H. Freedman and Frank Quinn. Topology of 4-Manifolds (PMS-39), Volume 39 . Princeton University Press (1990).
  For 2) also see appendix of https://math.berkeley.edu/~teichner/Papers/Alexander.pdf
Constructions of 4-manifolds
Main reference: Chapter 6 in [A] and references therein. An overview of the behaviour of SW invariants with respect to these operations is given in Chap. 13 of [A].
Small exotic manifolds
Main reference: Chapter 9 in [A] and references therein.
Corks, Anticorks & Plugs
Main reference: Chapter 10 in [A] and references therein.
Fintushel and Stern's reverse engineering
R. Fintushel and R.J. Stern, Surgery on nullhomologous tori, Geom. Topol. Monogr. 18 (2012) 61-81.
R. Fintushel, B.D. Park and R. J. Stern, Reverse engineering small 4-manifolds, Algebr. Geom. Topol. 8 (2007) 2103-2116.
Akhmedov and Park construction
A. Akhmedov and B.D. Park, Exotic 4-manifolds with small Euler characteristic, Invent. Math. 173 (2008) 209-223.
S. Akbulut, The Akhmedov-Park exotic CP2#3CP¯ 2, Adv. Math. 274 (2015) 928.947.

Contact

For more information about the seminar contact the organizer Roberto Ladu by sending an e-mail to r dot ladu19 at imperial dot ac dot uk