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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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