Topology - Core

Venue: Chern Lecture Hall

Class Timings: Tuesdays & Fridays from 9:45 AM -11:15 AM

First Meeting: 7 January 2025

Course Description: 

• Rapid review of relevant point-set topology.
• Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces.
• CW complexes, basics of homology and cohomology (singular and cellular); the Mayer-Vietoris sequence, excision, Künneth and universal coefficient theorems, isomorphism with de Rham cohomology.
• Cup and cap products, Poincaré duality, Alexander duality, Lefschetz fixed point theorem.
• Optional topics: Higher homotopy groups, Whitehead's theorem, fibrations and cofibrations, Freudenthal suspension theorem, Blakers-Massey theorem.

Course Outcomes:

• Develop a foundational understanding of key concepts in algebraic topology.
• Apply homology and cohomology theories to analyze the properties of topological spaces
• Apply representation theory to various problems.