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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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