Growth and Lie brackets in the homotopy Lie algebra (Q1605637)

Let \(X\) be a simply connected space of finite type and denote by \(L\) its rational homotopy Lie algebra \(\pi_*(\Omega X;\mathbb{Q})\). It is conjectured that the hypotheses ``\(\dim L=\infty\) and \(X\) finite dimensional'' imply that \(L\) contains a free Lie subalgebra on two generators. In the paper under review, the authors investigate this conjecture in several directions. For instance, they prove the existence of a finite sequence \(x_1,\dots,x_d\) of elements of \(L\) and of some integer \(N\) such that \([x_i,y]\neq 0\) for any \(y\in L\) of degree \(\geq N\). Another conjecture concerning the exponential growth of some dimensions of the \(L_i\) is also investigated. Similar results are also obtained for a local Noetherian commutative ring \(R\), of residual field \(k\), with \(UL= \text{Ext}_R (k,k)\).