Algebraic Topology

Prerequisite(s)：

Geometry and Topology, Abstract Algebra II

Course description:

This course is designed for third/fourth year undergraduate students and graduate students who intend to conduct research work in algebraic topology, differential geometry, algebraic geometry or closely related fields. The teaching objective is to endow students with the basic ideas and basic techniques in algebraic topology.

Learning objectives:

By the end of this course, students will be able to:

1) Understand the concept of singular homology and its various properties, know the basic examples: the homology of spheres and projective spaces.

2) Understand the concept of singular cohomology, know the basic examples such as the case for spheres and projective spaces.

3) Understand Poincare duality for closed oriented manifolds.

4) Master several techniques for computing (co)homology: long exact sequence of relative (co)homology, Excision, MV-sequences, CW-complexes, Kunneth formula, Universal coefficient theorem, Poincare duality.

5) The basic concept of fundamental groups and covering spaces.

Detailed topics covered:

1）Homology: singular chain complexes, homology groups/modules, relative singular chain complex, relative homology, homotopy invariance, long exact sequences, example: homology of spheres.

2) Computational tools: excision, MV-sequences, Eilenberg-Zilber equivalence, Kunneth's theorem, CW complexes,

3) Cohomology: singular cochain complexes, cohomology groups/modules, cup/cap products, the Universal coefficient theorem, local coefficient systems, manifolds, Poincare duality.

4) Fundamental groups and covering spaces: fundamental groups, deck transformations, Universal covering spaces, the Galois correspondence.