Universal Higher Extensions (11rit174)

Objectives

Let us now outline our project: {bf (I)}quad In one stream of development, one partner in this application, has just achieved an interpretation of group cohomology or Lie-algebra cohomology in terms of higher-dimensional central extensions, as developed by Rodelo-Van der Linden. It extends the classical interpretation of the second group cohomology in terms of equivalence classes of short exact sequences with central kernel. This development builds upon the notion of semi-abelian categories as in Janelidze-Marki-Tholen, and the concept of `higher central extension' developed within the framework of categorical Galois theory based on the work of Janelidze et. al. In addition, methods from the theory of simplicial groups are used.

{bf (II)}quad In a parallel and complementary stream of development, the second partner in this application has just achieved a proof of existence of universal $n$-step extensions of modules over an arbitrary unitary ring. This development involves certain higher torsion theories. It immediately has several applications ranging from
begin{itemize}
item a torsion theoretic conceptual hindsight explanation for existing computational results about the effect of plus constructions on the homotopy groups of a space, to
item identifying those groups, respectively Lie-algebras, which have a central $n$-extension in the sense of Rodelo-Van der Linden which is universal, to
item speculation about probable analogues in other non-abelian categories, notably the category of $Pi$-algebras.
end{itemize}

{bf Unanswered questions}quad While already a superficial interface between these two developments leads to exciting insights, a lot of questions remain unanswered (e.g. how about a constructive description of these abstractly existing universal central $n$-extensions?). Moreover, as hinted above, a number of further developments appear promising.

Thus it is natural for us to form a team with the objective to merge our complementary viewpoints and backgrounds on this subject and thus advance it as efficiently as possible.