Math 531

Math 531

Algebraic Topology

Professor Michael Sullivan

Spring 2016

The idea in algebraic topology is to associate to a topological space an algebraic object such as a group or ring. If two topological spaces are mapped to non isomorphic algebraic objects then they cannot be homeomorphic. Thus algebraic topology becomes a major tool for the study of topological spaces, especially manifolds and CW-complexes.

The main tools are the following.

Algebraic topology has applications in dynamical systems (including differential equations), mathematical physics, image processing and computer science. It has also evolved into an independent area of work within algebra.

As you might have summarized algebraic topology is a demanding topic. But this is an introductory course and I realize most students will be doing or beginning to do research in other fields. Thus, while the material is harder than that in 530, the amount the of homework is less and any tests will be take home. There will be an emphasis on connecting the geometric or visual understanding of spaces with the algebraic properties of their groups.


Instructor: Prof. Sullivan. Neckers 385.

Coordinates: T&Th 2:00-3:15, Engineering A 210. CRN=27364.

Material:We will finish some material on the fundamental group, Π1, from the end of Munkres' textbook Topology. Then we will do a little on the higher homotopy groups, Πn, using Hatcher's textbook Algebraic Topology (free online). Most of the course will be on homology using Munkres' textbook Elements of Algebraic Topology. (This is the only book you will need to buy.) We will touch on cohomology toward the end.

Grades: Homework will be collected about every other week. There will be a takehome final. Grades will be 50% homework, 50% final.

Prerequisites: A basic topology course such as Math 530 and some knowledge of group theory and linear algebra at the undergrad level.


  • Tietze Transformations
  • Connected Sums of 3-Manifolds
  • Knot Groups