242552 Spectral Sequences and Topological Applications (S) (SoSe 2021)

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Requirements for participation, required level

Spectral sequences are a fundamental feature in the toolkit of any algebraist or topologist. Essentially they should be thought of as machines that convert local algebraic information into global homological information. As a basic example, the skeletal filtration of a CW complex gives rise to a filtration of its homology which allows for effective computation. While this is elementary, spectral sequences become indispensible when the same methods are used to compute the homology of more complicated geometric objects like fibre bundles.

Indeed, this will be one of our primary motivations. In particular we will use the Leray-Serre spectral sequence to study the characteristic classes that classify vector bundles and other algebraic objects. We will also find application for it in the study of the Steenrod algebra, which has a ubiquitous presence not only in algebraic topology, but also in areas like group cohomology.

We will follow closely J. McCleary's book "A User's Guide to Spectral Sequences". Depending on the audience there will be scope to incorporate more or less material relating to rings and group cohomology. A basic requirement for the course is (singular) homology and basic homological algebra. The student should be familiar with the notions of exactness, homology, and functors like Ext and Tor. We assume no special knowledge of rings, nor any particular knowledge of topology past what is offered in a one semester Bachelor course.

The course will be taught in English by myself. For further information feel free to get in touch.

Bibliography

J. McCleary - A User's Guide to Spectral Sequences.
J. Rotman - An Introduction to Homological Algebra.
C. Weibel - An Introduction to Homological Algebra.