分野別セミナー

The homotopy theory associated to classifying spaces of finite groups and compact Lie groups has been a subject of considerable interest to homotopy theorist for several decades. In this talk I will present the foundations of a theory due to C. Broto, B. Oliver and myself, called p-local group theory. A p-local group is an algebraic object which is modelled on the homotopy theory of a p-completed classifying space of a finite group, or more generally a compact Lie group. In particular p-local groups have classifying spaces. The family of spaces which arises in this way shares many of the properties one finds in p-completed classifying spaces of genuine groups, but there are many examples of p-local groups which are exotic. The theory is also closely related to modular representation theory, particularly the fundamental concept of a fusion system. After a survey of the main definitions and results in the subject I will also attempt to give some recent advances.