The cohomology of homotopy categories and the general linear group (Q913942)

The authors define cohomology groups of a very general kind for small categories and consider their properties. The second cohomology groups classify certain kinds of extensions. If \({\mathbb{C}}\) is a suitable subcategory of the homotopy category, then it has a canonical 3- dimensional cohomology class \(<{\mathbb{C}}>\), which acts as a universal Toda bracket. Several examples are given. In particular, there is an elegant description of Igusa's associativity class \(\chi\) (1) in the cohomology of \(GL_ n({\mathbb{Z}})\). It is got by considering wedges of spheres and automorphisms of nil(2)-groups (groups of the form F/[F,[F,F]] with F a free group).