- Dozent: Thomas Schick
- Time: Mo,Thu, 8:15-9:55
- exercise class: to be determined
- Room: HS 3
- Kontact: schick@math.uni-goettingen.de, Tel. 39-7799
- credit: oral exam after the semester

The forth part of the course on algebraic topology gives a systematic study of cohomology theories via the so-called spectra.

The spectra in algebraic topology are a basic tool in homotopy theory. The short description is: spectra are a generalization of topological spaces, where the suspension construction has an inverse: every spectrum is a suspension of some other spectrum (at least upto homotopy equivalence). As a consequence, spectra often have homotopy groups in negative degrees.

It turns out that to every spectrum one can associated a generalized (co)homology theory, and that, on the other hand, every such is represented by a spectrum. Transformations of generalized cohomology theories correpond to maps of spectra.

In the course we will

- introduce the category of spectra (objects and maps), -construct the associated (co)homology theory,
- learn about the structure theory (e.g. fibration and cofibrations of spectra, assoziated decomposition of cohomology theories)
- describe some of the most important examples: sphere spectrum, Eilenberg-MacLane spectra, Thom spectra corresponding to bordism theories, K-theory spectrum.

Prerequisites

homology and cohomology, homotopy groups, fibrations. Not required: knowledge of spectral sequences.

Literature (among others):

- Yuli Rudyak: On Thom spectra, orientability, and cobordism
- Kochman: Bordism, stable homotopy and Adams spectal sequence
- Paul Selick: Introduction to homotopy theory
- Davis, Kirk: Lecture notes in algebraic topology
- Robert Switzer: Algebraic topology: homotopy and homology