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Co-\(H\)-structures on Moore spaces of type \((G,2)\). (English) Zbl 0829.55006
Let \({\mathcal E}_* (X)\) be the kernel of \({\mathcal E} (X) \to \text{aut} H_* (X)\) and \({\mathcal E}_\# (X)\) be the kernel of \({\mathcal E} (X) \to \text{aut} \Pi_* (X)\) where \({\mathcal E} (X)\) is the group of homotopy classes of self-homotopy equivalences of \(X\), and let \({\mathcal E}_{* \#} (X) = {\mathcal E}_* (X) \cap {\mathcal E}_\# (X)\). The authors establish some general results under which \({\mathcal E}_* (X)\), \({\mathcal E}_\# (X)\) or \({\mathcal E}_{*\#} (X)\) is finite or infinite. They then give specific finiteness results where \(X\) is a homogeneous space of the form \(X = U(n)/(U(n_1) \times \cdots \times U(n_k))\), and where \(X\) is a product of spheres. Their method is to consider the minimal model and determine finiteness for the corresponding kernels. They discuss obstruction theory for homotopy of homomorphisms \(f : {\mathcal M} \to {\mathcal N}\) from a 2-stage minimal DGA to a general minimal DGA. For the nilpotent spaces of finite type which they consider, \([X,Y] \to \text{hom} (H^* (Y; \mathbb{Q}),\), \(H^* (X; \mathbb{Q}))\) is finite-to-one.

MSC:
55P62 Rational homotopy theory
57T15 Homology and cohomology of homogeneous spaces of Lie groups
55P10 Homotopy equivalences in algebraic topology
55P40 Suspensions
55P45 \(H\)-spaces and duals
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