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Homotopy groups of certain highly connected manifolds via loop space homology. (English) Zbl 1422.55017
Let $$n\geq 2$$ and $$d$$ be positive integers such that $$d\leq 3n-2$$. In this paper the authors study the homotopy types of $$(n-1)$$-connected $$d$$-dimensional closed manifolds. In particular, when $$\dim H^*(M;\mathbb{Q})>4$$ they show that the $$p$$-local homotopy groups of $$M$$ can be determined by the dimension of the space of indecomposable elements in the cohomology ring $$H^*(M;\mathbb{Q})$$ and prove that these $$p$$-local homotopy groups can be expressed as the finite $$p$$-local homotopy groups of spheres. Their proof is based on a careful analysis of the loop homology $$H_*(\Omega M;\mathbb{Q})$$.
##### MSC:
 55P15 Classification of homotopy type 55P35 Loop spaces 55Q52 Homotopy groups of special spaces 16S37 Quadratic and Koszul algebras 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010)
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