A counterexample to the Lemaire-Sigrist conjecture (Q1292677)

Recall that the Lyusternik-Shnirel'man category of a space \(X\), \(\text{cat} X\), is the least integer \(n\) such that \(X\) can be covered by \(n+1\) open sets each of them contractible in \(X\). The integer \(\text{cat} X\) is clearly less than or equal to the cone length of \(X\), \(\text{cl}(X)\), that is the least integer \(n\) for which there are cofibration sequences \(\Sigma^i Z_i \to X_i\to X_{i+1}\), \(0\leq i<n\), where \(X_n\) has the same homotopy type as \(X\) and \(X_0\) is contractible. O. Cornea proved that \(\text{cat} X\leq\text{cl}(X) \leq\text{cat} X+1\), while Lemaire and Sigrist conjecture that for a simply connected rational space \(\text{cat} X=\text{cl}(X)\). In this paper the author gives the first example of a rational space with \(\text{cl} (X)=\text{cat} X +1\). The proof is a very nice use of differential graded Lie models for rational space. The category of the space \(X\) is three which is the minimal possible value for such an example, because for spaces \(Y\) with \(\text{cat} Y\leq 2\), we have \(\text{cat} Y=\text{cl}(Y)\).