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Homotopy groups of highly connected manifolds. (English) Zbl 1433.55005
The authors prove the following result (Theorem A). Let $$M$$ be a closed $$(n-1)$$-connected $$2n$$-manifold with $$n\ge 2$$ and $$n$$th Betti number $$r\ge 2$$. (a) The homotopy groups of $$M$$ can be expressed as a direct sum of homotopy groups of spheres. (b) The homotopy groups of $$M$$ are determined by $$r$$, i.e., the number of factors $$\pi_jS^\ell$$ occurring in the homotopy group $$\pi_jM$$ is a function of $$\ell,r$$ and $$n$$ only. The case when the $$n$$th Betti number is 1 turns out to be different. As a consequence of Theorem A, a Moore conjecture, regarding the relationship between the rational and torsion homotopy groups of finite simply connected CW-complexes, is established in this particular case. Similar results were obtained by P. Beben and S. Theriault [Adv. Math. 262, 213–238 (2014; Zbl 1296.55012)] by different methods, $$n=4$$ and $$n=8$$ excepted. The primary techniques used in the proof of this theorem are quadratic algebras, Koszul duality of associative algebras and quadratic Lie algebras. The homology of the loop space $$\Omega M$$ which is a tensor algebra modulo a relation corresponding to the attaching map of the top cell, is calculated via a quadratic Lie algebra. These techniques are also used to thoroughly analyze homotopy groups of a CW complex $$X=(\bigvee_rS^n) \cup e^{2n}$$ obtained by attaching a $$2n$$-dimensional cell to a finite wedge of $$n$$-dimensional spheres (Theorem C) when the intersection form of $$X$$ has rank at least 2 in Theorem A.

##### MSC:
 55P35 Loop spaces 55Q52 Homotopy groups of special spaces 57N65 Algebraic topology of manifolds 16S37 Quadratic and Koszul algebras
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