Lecture course "`Algebraic Topology IV, SoSe 14"'

• 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
Contents of the course:

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