The 10/8^ths Theorem

The details: this is the webpage for a learning seminar on Furuta's paper [Fur01], which uses Seiberg-Witten theory and equivariant K-theory to prove the 10/8^ths theorem on the intersection forms of spin 4-manifolds. This seminar is organized by Arun Debray and Riccardo Pedrotti.

Logistics:

• Where and when: Mondays 10AM in PMA 9.166.
• Lecture notes can be found here.

Lecture schedule:

Planning meeting and overview

Arun and Riccardo, January 23

Compactness of the Seiberg-Witten moduli space

Kai and Riccardo, January 28 and February 4

The Pin₂^–-symmetry of the Seiberg-Witten equations

Arun, February 11

Finite-dimensional approximation

Cameron and Leon, February 18 and 25

Equivariant K-theory

Richard, March 4

Proof of the 10/8^ths theorem

Arun, March 11

Z/2^p-actions on spin 4-manifolds

Arun and Jeffrey, March 25 and April 1

Bauer-Furuta invariants

Jeffrey, Leon, and Riccardo, April 15, April 22, April 29

Families Seiberg-Witten invariants

Arun, May 6

References:

[BF03]

S. Bauer and M. Furuta. A stable cohomotopy refinement of Seiberg-Witten invariants: I. Invent. math. (2004) 155: 1.

[Bau03]

S. Bauer. A stable cohomotopy refinement of Seiberg-Witten invariants: II. Invent. math. (2004) 155: 21.

[Bry98]

Jim Bryan, Seiberg-Witten theory and Z/2^p actions on spin 4-manifolds. Math. Res. Lett., 5(2):165–183, 1998.

[Fur01]

M. Furuta. Monopole equation and the 11/8-conjecture. Math. Res. Lett., 8(3):279–291, 2001.

Related links:

[DK19]

David Baraglia and Hokuto Konno. On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds.

[HLSX18]

Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu. Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant.

[LL01]

Tian-Jun Li and Ai-Ko Liu. Family Seiberg–Witten invariants and wall crossing formulas. Comm. Analysis and Geometry, Volum 9, Number 4, 777-823, 2001.

[Nak03]

Nobuhiko Nakamura, The Seiberg-Witten equations for families and diffeomorphisms of 4-manifolds. Asian J. Math. 7 (2003) 133-138, Correction: Asian J. Math. 9 (2005) 185-186.

[Szy10]

Markus Szymik. Characteristic cohomotopy classes for families of 4-manifolds. Forum Mathematicum, 22.3 (2010): 509-523.

[Szy13]

Markus Szymik, The stable homotopy theory of vortices on Riemann surfaces.