Math 564/565/566
Algebraic Topology
Ethan Devinatz
Autumn/Winter/Spring 2002-2003, Monday/Wednesday/Friday 2:30-3:20
The first two quarters of this sequence will be a fairly standard
introductory algebraic topology course. Topics to be covered include:
simplicial complexes, CW complexes, singular and cellular homology
and cohomology, universal coefficient theorems and Künneth formulas,
the Eilenberg-Steenrod axioms, Poincaré duality. In the third
quarter, we will begin studying homotopy theory. We hope to discuss
(although this is probably too ambitious): higher homotopy groups,
exact sequences of mapping functors, the Hurewicz theorem, homotopy theory
of CW complexes, obstruction theory, Eilenberg-MacLane spaces.
Prerequisites:
- From algebra: a little group theory, knowledge of structure of
finitely generated modules over a principal ideal domain. We will
develop the required homological algebra as needed.
- From point-set topology: a working knowldege of point-set topology;
e.g., compact Hausdorff spaces, connected spaces.
- Miscellaneous: definition of a manifold, knowledge of the fundamental
group and the classification of covering spaces. (These topics are
covered in the first year manifolds course.)
Texts: The main text for the first two quarters will be Homology
theory by J.W. Vick. I will also be using Spanier's classic text
Algebraic topology as a supplement, although neither text will be
followed slavishly. I have not yet decided on a text for the third
quarter.