MAT 539
Algebraic Topology

Instructor    Sorin Popescu   (office: Math 4-119, tel. 632-8358, e-mail


A basic introduction to geometry/topology, such as MAT 530 and MAT 531. Thus prior exposure to basic point set topology, homotopy, fundamental group, covering spaces is assumed, as well as some acquaintance with differentiable manifolds and maps, differential forms, the Poincaré Lemma, integration and volume on manifolds, Stokes' Theorem. We will briefly review some of this material in the first week of classes.


Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu, GTM 82, Springer Verlag 1982.

The guiding principle of the book is to use differential forms and in fact the de Rham theory of differential forms as a prototype of all cohomology thus enabling an easier access to the machineries of algebraic topology in the realm of smooth manifolds. The material is structured around four core sections: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes, and includes also some applications to homotopy theory.

Other recommended texts:

  • Algebraic Topology: A first Course, W. Fulton, GTM 153, Springer Verlag 1995
  • Topology from the Differentiable Viewpoint, J. Milnor, U. of Virginia Press 1965
  • Algebraic Topology, A. Hatcher (on-line), Cambridge University Press, to appear
  • Characteristic classes, J. Milnor and J. Stasheff, Princeton University Press 1974

Course description

The book contains more material than can be resonably covered in a one-semester course. We will hopefully cover the following sections:

Homework & Exams

I will assign problems in each lecture, ranging in difficulty from routine to more challenging. There will be also a take-home midterm and a final exam. Course grades will be based on these problems (and any other participation); solving at least half of them will be considered a perfect score.


Here are some pointers to software that may be used to visualize topological objects:

Links & 3D-models

History of topology:    The Koenigsberg bridges

Topological zoo:    Crosscap

Art & Topology:    Trefoil Knot

Archives:    Figure Eight Knot

Fun:    Klein Bottle by Michael James Grady


Sorin Popescu