A minimum dimensional counterexample to Ganeaʹs conjecture

A 7-dimensional CW-complex having Lusternik–Schnirelmann category equal to 2 is constructed. Using a divisibility phenomenon for Hopf invariants, it is proved that the Cartesian product of the constructed complex with a sphere of sufficiently large dimension also has category 2. This space hence constitutes the minimum dimensional known counterexample to Ganeaʹs conjecture on the Lusternik–Schnirelmann category of spaces.