# Stable Homotopy Theory

#### Trinity Term 2012. Class TR9:30-11 in SGSR1; discussion session F13-14 in SGT14.

##### Course description

The aim of the course will be to develop a general theoretical and computational familiarity with Adams spectral sequences for the stable homotopy groups of spheres. We will begin by discussing various background topics, including spectra, the Steenrod algebra, localization and completion, and the relevant homological algebra. Thusly equipped we will construct the Adams spectral sequence, identify its E_2 term, and perform some calculations. We will then step back and study formal groups, complex-oriented cohomology theories, and Hopf algebroids, with a view toward understanding the context of the Adams-Novikov spectral sequence. At this point, which may begin a continuation course in Michelmas, we can investigate global properties of stable homotopy such as nilpotence and periodicity, and begin studying the chromatic picture.

Roughly speaking, I will assume familiarity with the content of Hatcher, `additional topics’ excepted. The course will begin with a discussion of spectra. Needless to say, students and others in geometry, representation theory, algebra, and other disciplines, are all welcome.

##### Contact information

Christopher Douglas

Office: SGS7

Email: cdouglas at maths