Math 442 - Introductory Topology II

3 credits. Offered in most Spring semesters. Official course information. Prerequisites: Math 441.

Description

The main idea of Math 442 is to build a foundation for algebraic topology, the study of topological spaces using methods and ideas from abstract algebra. This works by assigning certain abstract algebraic structures (like groups, rings, or modules) to topological spaces in an invariant way (meaning that two spaces with different invariants are definitely different topological spaces). This class will introduce the theory of how this works and present some neat applications of this theory.

Topics covered

Topics covered in Math 442 usually include:

  • Homotopy and homotopy equivalence of spaces
  • CW complexes and various other operations on topological spaces (quotients, wedge sum, suspension, etc.)
  • Fundamental group: applications (fundamental theorem of algebra), and methods of computation (van Kampen’s theorem)
  • Covering spaces: examples, their classification, and connection to the fundamental group
  • Homology theory: the underlying algebraic theory (homological algebra), applications (e.g. the famous “hairy ball theorem”, etc.)

Note that the exact list of topics may vary by instructor and semester. Additional topics may include basic objects of algebraic topology, such as cohomology theory, the elementary theory of vector bundles and K-theory, and elementary knot theory.

Comments on prerequisites

Most of the advanced algebra required for this class is self-contained, i.e. it will be taught as it is needed and comes up. Nevertheless, it can be helpful to have some algebra background or read up on some algebra (in particular, basic group theory).

Resources

Textbook

The usual textbook for the class is Algebraic Topology by Allen Hatcher, which is freely available on the author’s website at https://pi.math.cornell.edu/~hatcher/AT/ATpage.html.

Additional materials

Some past course webpages are available online.