Math285Z: Sutured 3-manifolds and Floer homology

Floer homology is a generalization of Morse homology to infinite-dimensional spaces and is an important tool in low-dimensional topology and symplectic geometry. This course introduces the basic theory of sutured 3-manifolds and Floer homology, including Heegaard Floer homology, monopole Floer homology, instanton Floer homology, and more.

Course Information

Course Name: Sutured 3-manifolds and Floer homology

Course Number: Math285Z

Instructor: Fan Ye

Course Content:

This course covers the following main topics:

• Basic theory of sutured 3-manifolds
• Heegaard Floer homology
• Monopole Floer homology
• Instanton Floer homology
• Knot Floer homology
• Applications of Floer homology

Textbooks:

Most references are papers. Notes will be posted on the course page after the classes. The names in the references denote the corresponding documents.

Tentative Course Outline:

• Weeks 1-2 (Jan. 22-31): Introduction to sutured manifolds. Ref: Lipshitz, Juhász
• Weeks 3-4 (Feb. 5-14): Sutured Heegaard Floer homology. Ref: Lipshitz, Juhász, Manolescu
• Weeks 5-6 (Feb. 19-28, with a holiday on Feb. 19): Sutured monopole Floer homology. Ref: Lin, Kronheimer-Mrowka
• Weeks 7-9 (Mar. 4-20, with spring break during Mar. 9-17): Sutured instanton Floer homology. Ref: Wang, Kronheimer-Mrowka
• Weeks 10-11 (Mar. 25-Apr. 3): Contact elements and surgery exact triangles. Ref: Sahamie, Oszváth-Szabó
• Weeks 12-14 (Apr. 8-24): Towards isomorphism of Floer homology. Ref: TBD

Prerequisites:

• Differential geometry, especially principal bundle. Ref: a) Differential Geometry: Bundles, Connections, Metrics and Curvature, Chapters 1-16, Clifford H. Taubes; b) An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby; c) Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner.
• Algebraic topology, especially homology theory, characteristic classes, and spectral sequences. Ref: a) Algebraic Topology; b) Vector Bundles & K-Theory; both by Allen Hatcher; c) Differential Forms in Algebraic Topology, Chapters 4-5, Raoul Bott, Loring W. Tu.
• 3-manifold topology. Ref: Notes on Basic 3-Manifold Topology, Allen Hatcher.

Course History

Note: This course information is compiled based on related course content from Harvard University.