Homotopy patterns in group theory

  • Roman Mikhailov

    Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178, Russia, and St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, Saint Petersburg, 191023, Russia
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This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of nonadditive functors, homotopy groups of spheres, group homology, etc., appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.