Faithful actions on commutative differential graded algebras and the group isomorphism problem (Q1790256)

The realizability problem for abstract groups, proposed by \textit{D. W. Kahn} [Math. Ann. 220, 37--46 (1976; Zbl 0305.55016)] asks for a characterization of groups that appear as the group of self-homotopy equivalences of simply-connected spaces. The authors present the realizability problem for group actions which is an extension of Kahn's realizability problem. Let the pair \((G,M)_R\) denote a group \(G\) acting on an \(R\)-module \(M\), \(R\) a unitary ring. They ask for the existence of an \(R\)-local space \(X\) such that \((\mathcal{E}(X), \pi_k(X))_R\) is equivalent, in a natural way, to \((G,M)_R\), for some \(k\ge 2\), where \(\mathcal{E}(X)\) denotes the group of homotopy classes of self-homotopy equivalences of \(X\) acting by composition on the homotopy groups \(\pi_k(X)\). It is proved that if \(G\) is finite and acts faithfully on a finitely generated \(\mathbb{Q}\)-module \(M\) then there exist infinitely many rational spaces realizing \((G, M)_\mathbb{Q}\). The alternate approach via invariant theory also provides motivation and guidance to tackle Kahn's problem in the case of infinite groups. The authors realize a large class of orthogonal groups strictly containing the class of finite groups. Then, their result [Acta Math. 213, No. 1, 49--62 (2014; Zbl 1308.55005)] is improved by enlarging the class of groups that are known to be realizable in the classical Kahn's sense.