Homotopy groups of highly connected manifolds (Q1785790)

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 \textit{P. Beben} and \textit{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.