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.
Faithful actions on commutative differential graded algebras and the group isomorphism problem
scientific article; zbMATH DE number 6353666
FAITHFUL ACTIONS ON COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS AND THE GROUP ISOMORPHISM PROBLEM
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