On the deviations of a \(\mathbb{Q}\)-algebra
The study of the rational homotopy type of 1-connected spaces is related to the study of local rings. From this point of view, to the graded commutative algebra \(H = H^*(X;\mathbb{Q})\) is naturally associated the series \[ \sum^ \infty_{j=0}\dim\text{Tor}^ H_ j(\mathbb{Q},\mathbb{Q})t^ j = \prod^ \infty_{i=0}{(1+t^{2i+1})^{\varepsilon_{2i}}\over (1-t^{2i+2})^{\varepsilon_{2i+1}}} \] and the Gulliksen-Levin conjecture translates to: If \(H\) is Noetherian and \(\varepsilon_ i = 0\) for some \(i \geq 2\) then all \(\varepsilon_ i\) are zero for \(i \geq 2\). The author proves this conjecture when \(H\) has finite cuplength.
rational homotopy type of 1-connected spaces
scientific article; zbMATH DE number 167299