A COUNTEREXAMPLE TO THE LEMAIRE–SIGRIST CONJECTURE

We give an example of a rational space X such that its cone-length, Cl(X), is equal to Cl(X∨S)+1, where S is some rational sphere. In particular, the Lusternik–Schnirelmann category of X, cat(X), is strictly smaller than the strong Lusternik–Schnirelmann category of X, Cat(X). This is the first example of a rational space with this property and it contradicts the Lemaire–Sigrist conjecture.