Algebraic Topology by Allen Hatcher, Cambridge Univ & Homotopical Topology by Fomenko-Fuchs

Graduate standing or consent of instructor

Homotopy, fundamental group, covering spaces, higher homotopy groups

• (1 lecture) Examples and constructions of topological spaces: spheres, projective spaces, products, wedge, suspension, CW complexes and decompositions [Hatcher Ch 0]

• (2 lectures) Homotopy: definitions and examples of homotopy, homotopy equivalence, retracts, deformation retracts, contractible spaces, π0 and path connectedness, homotopy extension, homotopy invariance under quotient by a contractible subspace [Hatcher Ch 0]

• (2 lectures) Fundamental group basics: definition of π1, induced maps, homotopy invariance, group structure, changing base points [Hatcher 1.1]

• (1-2 lectures) Basic examples of computing π1: contractible spaces, S1, products [Hatcher 1.1]

• (3 lectures) Seifert van Kampen: theorem statement, applications in examples, proof of theorem [Hatcher 1.2]

• (1-2 lectures) Basics of covering spaces: definition and examples, homotopy lifting, injectivity of p∗, deck transformations, regular coverings [Hatcher 1.3]

• (3 lectures) Classification of covering spaces: statement and proof, universal covers [Hatcher 1.3]

• (2 lectures) Basics of higher homotopy groups: definitions, basic examples, group structure, changing basepoints, covering maps induce isomorphisms [Hatcher 4.1]

• (2-3 lectures) Relative homotopy groups of pairs: definition, long exact sequence [Hatcher 4.1]

• (3 lectures) Fiber bundles: Definitions, examples, homotopy extension lifting, Serre fibrations, homotopy long exact sequence and computational applications in examples [Fomenko-Fuchs Lecture 9]

• (if time allows) Homotopy groups and cell complexes, Freudenthal suspension [Fomenko-Fuchs Lecture 10-11]