Rational nilpotent groups as subgroups of self-homotopy equivalences (Q2752244)

Some instances of the realization problem for subgroups of homotopy self-equivalences are considered. Given a space \(X\), the set of homotopy classes of homotopy equivalences from \(X\) to itself form a group with respect to composition. It is called the group of self-homotopy equivalences of \(X\), and denoted \({\mathcal E}(X)\). The realization problem is one of the main open problems concerning groups of self-homotopy equivalences: given a group \(G\), is there a space \(X\), such that \(G\cong {\mathcal E}(X)\)? There are several partial answers to this problem but the complete solution is still unknown. The same question is studied for certain subgroups of \({\mathcal E}(X)\) as well, notably for \({\mathcal E}_\sharp(X)\) and \({\mathcal E}_\ast(X)\), consisting of classes that induce identity homomorphisms on homotopy and homology groups, respectively (up to the dimension of \(X\)). For these, the realization problem seems to be easier, due to the somewhat simpler structure of the groups in question. Indeed, by a classical theorem of \textit{E. Dror} and \textit{A. Zabrodsky} [Topology 18, 187-197 (1979; Zbl 0417.55008)], \({\mathcal E}_\sharp(X)\) and \({\mathcal E}_\ast(X)\) are nilpotent groups. Moreover, by results of \textit{K.-I. Maruyama} [Pac. J. Math. 136, No. 2, 293-301 (1989; Zbl 0673.55006); Math. Proc. Camb. Philos. Soc. 108, No. 2, 291-297 (1990; Zbl 0718.55006)] these groups can be localized, so in particular we have that \({\mathcal E}_\sharp(X_{\mathbb Q})\cong {\mathcal E}_\sharp(X)_{\mathbb Q}\), and \({\mathcal E}_\ast(X_{\mathbb Q})\cong {\mathcal E}_\ast(X)_{\mathbb Q}\) (where \((-)_{\mathbb Q}\) denotes respectively the Sullivan rationalization of spaces and the Malcev completion of nilpotent groups). The main achievement of the paper is the explicit construction of spaces for which the groups \({\mathcal E}_\sharp(X)\) and \({\mathcal E}_\ast(X)\) are isomorphic to Malcev completions of nilpotent groups of nilpotency \(\leq 2\), and Hirsch rank \(\leq 6\). For \({\mathcal E}_\sharp\), the spaces in question are rationalizations of products of spheres, complex projective spaces and certain homogeneous spaces, while for \({\mathcal E}_\ast\) the examples are rationalizations of wedges of spheres and cofibres of maps between them. The techniques of proof are quite involved: they consist of some heavy computations, based on the theory of Sullivan and Quillen minimal models, obstruction theory for rational homotopy equivalences, due to Arkowitz, Lupton, Halperin and Stasheff, and the classification of nilpotent groups of low rank by Grunewald, Scharlau, Segal and O'Haloran. Apart from the precise results, the main interest of the paper is that it contains, albeit limited in scope, the first systematic treatment of the realization problem for a class of non-commutative groups.