Rational homotopy groups and Koszul algebras
If \(A\) is a graded algebra, a rescaling of \(A\) is the graded algebra \(A[k]\) with \(A[k]^{q(2k+1)} =A^q\) and \(A[k]^p=0\) otherwise. Here, the authors study different properties linking two CW-spaces \(X\) and \(Y\) \((X\) connected and \(Y\) simply connected) of finite type, such that \(H^*(Y; \mathbb{Q})\) is a rescaling of \(H^*(X;\mathbb{Q})\). For instance, they prove: -- if \(H^*(X;\mathbb{Q})\) is a Koszul algebra (which means \(\text{Tor}_{p,q}^{H^* (X;\mathbb{Q})}=0\) if \(p\neq q)\) then \(\pi_*(\Omega Y)\otimes \mathbb{Q}\) is a rescaling of \(\text{gr} (\pi_1(X)) \otimes \mathbb{Q}\), -- if \(H^*(X;\mathbb{Q})\) is a Koszul algebra and \(X\) is formal, then there exist isomorphisms of groups between the Malcev completion of \(\pi_1 (X)\), the completion of \([\Omega S^{2k+1},\Omega Y]\) and the group of coalgebra maps from \(H_*(\Omega S^{2k+1}; \mathbb{Q})\) to \(H_*(\Omega Y;\mathbb{Q})\). Applications to hyperplane arrangements are given.
scientific article; zbMATH DE number 1777364
scientific article