Homotopikategorien

Forkunnskaper : MAT242.

Beskrivelse : In this project we shall study the homotopy category of topological spaces. The homotopy category is what one obtains if instead of looking at continuous maps between topological spaces one considers homotopy classes of continuous maps. This category plays a fundamental role in algebraic topology since many questions about topological spaces can be answered by analysing such homotopy classes of maps.

There are two types of continuous maps that play a special role in the study of the homotopy category: the fibrations and the cofibrations. For example, any covering map is a fibration and any “nice inclusion” is a cofibration. The first part of the project is to understand these types of maps and to analyse various examples.

The properties of the fibrations and the cofibrations are formalised in the general notion of a model category. The second part of the project is to understand the notion of a model category and to analyse how the homotopy category of topological spaces fits into this general setting.

Referanser:

[1] W. G. Dwyer and J. Spalinsky, Homotopy theories and model categories, Online version.

[2] A. Strøm, The homotopy category is a homotopy category, Arch. Math. (Basel) 23, (1972), 435–441. (doi:10.1007/BF01304912).