Math231A: Algebraic Topology

Algebraic topology studies the properties of topological spaces through algebraic methods and is an important theoretical foundation for Floer homology. This course introduces basic concepts such as homology groups and cohomology, as well as important computational tools such as CW complexes, Mayer-Vietoris sequences, and universal coefficient theorems.

Course Information

Course Name: Algebraic Topology

Course Number: Math231A

Instructor: Fan Ye

Course Content:

This course covers the following main topics:

• Homology groups (singular homology, simplicial homology)
• Cohomology groups
• CW complexes
• Mayer-Vietoris sequences
• Basics of homological algebra

Textbooks:

Main References:

• Lectures on Algebraic Topology, Chapters 1-3, Haynes Miller
• Algebraic Topology, Chapters 2-3, Allen Hatcher

Other suggested references: W. Fulton: Algebraic Topology; E. Spanier: Algebraic Topology; Greenberg and Harper: Algebraic Topology: A First Course

Tentative Course Outline:

• Weeks 1-4: Singular Homology
• Weeks 5-8: Computational methods
• Weeks 9-14: Cohomology and duality

Prerequisites: An undergraduate-level understanding of topology. An undergraduate may look at Basic Topology by M. A. Armstrong.

Course History

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