Synopsis:

• This will be an introductory course in homotopy theory. We will begin by studying spaces which decompose as a product Y x Z. Next, we will consider ''twisted'' products, otherwise known as bundles. We will then consider the more general notion of a fibration, defined in terms of the homotopy lifting property. Cohomology will be a primary tool for studying fibrations, and this will lead us to consider spectral sequences, a tool for computing algebraic invariants like cohomology. Time permitting, towards the end of the course we will compute some homotopy groups of spheres.

Texts:

• Algebraic Topology, by Allen Hatcher. Some of the topics we will discuss are covered in Chapter 4 and in appendices.
• Differential Forms in Algebraic Topology, by Bott and Tu (Springer GTM 82). Especially Chapter 3.
• Cohomology Operations and Applications in Homotopy Theory, by Mosher and Tangora (Dover).

Homework: Homework exercises will be assigned occasionally to solidify the material in the lectures. You are strongly encouraged to work in groups on the homework, but you must write up your solutions independently.