In 1994, Witten [12] introduced a non-linear partial differential equation on a 4-manifold, called the Seiberg-Witten equations today. This PDE has brought significant progresses in 4-dimensional topology and geometry. In this series of lectures, I shall start with the basics of Seiberg-Witten theory and survey some of rather recent developments in Seiberg-Witten theory.

The first lecture is devoted to explain the foundation of Seiberg-Witten theory for closed 4-manifolds and some of classical major applications of it. Most results in this first lecture have been obtained in the last century, but they are explained from a relatively modern viewpoint: more concretely, my exposition shall be based on the technique of finite-dimensional approximations of the Seiberg-Witten equations, along Furuta [6] and Bauer-Furuta [5].

In the second lecture, I present what one can say about the diffeomorphism groups of 4-manifolds using Seiberg-Witten theory. This lecture contains much recent study on Seiberg-Witten theory for families of 4-manifolds. I shall explain (perhaps some of) results in [1, 2, 3, 4, 7, 8, 11] of this kind.

The third lecture will be a crash course in Seiberg-Witten Floer homotopy theory, started by Manolescu [10]. Amongst various versions of Floer theory known in low dimensional topology, this theory has a big advantage that one may consider any kind of generalized cohomology theory, such as K-theory. I shall sketch Manolescu's construction of Seiberg-Witten Floer homotopy theory, and explain a recent application in our paper [9] of this framework to knot theory.

Prerequisite:

No specific advanced knowledge is needed, but it would be helpful to be familiar with the basics of both of topology and differential geometry.

References:

[1] David Baraglia, Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants. Algebr. Geom. Topol. 21 (2021), no. 1, 317–349.

[2] David Baraglia and Hokuto Konno, A gluing formula for families Seiberg-Witten invariants. Geom. Topol. 24 (2020), no. 3, 1381–1456.

[3] David Baraglia and Hokuto Konno, On the Bauer-Furuta and Seiberg-Witten invariants of families of 4-manifolds, arXiv:1903.01649, to appear in J. Topol.

[4] David Baraglia and Hokuto Konno, A note on the Nielsen realization problem for K3 surfaces, arXiv:1908.03970, to appear in Proc. Amer. Math. Soc.

[5] Stefan Bauer and Mikio Furuta, A stable cohomotopy refinement of Seiberg-Witten invariants. I, Invent. Math. 155 (2004), no. 1, 1–19. 

[6] Mikio Furuta, Monopole equation and the 11/8-conjecture, Math. Res. Lett. 8 (2001), no. 3, 279–291.

[7] Tsuyoshi Kato, Hokuto Konno, and Nobuhiro Nakamura, Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds. Compos. Math. 157 (2021), no. 4, 770–808.

[8] Hokuto Konno, Masaki Taniguchi, The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary, arXiv:2010.00340

[9] Hokuto Konno, Jin Miyazawa, and Masaki Taniguchi, Involutions, knots, and Floer K-theory, arXiv:2110.09258

[10] Ciprian Manolescu, Seiberg-Witten-Floer stable homotopy type of three-manifolds with b1=0. Geom. Topol. 7 (2003), 889–932.

[11] Daniel Ruberman, An obstruction to smooth isotopy in dimension 4. Math. Res. Lett. 5 (1998), no. 6, 743–758.

[12] Edward Witten, Monopoles and four-manifolds. Math. Res. Lett. 1 (6):769–796, 1994.

Notes:

Lecture 1 2022.03.16.pdf  Lecture 2 2022.03.24.pdf  Lecture 3 2022.03.28.pdf