The Lusternik-Schnirelmann category of a topological space X
is a measure of the complexity of the space. When X is a manifold it
gives a lower bound for the number of critical points of a smooth
function on X. Ganea conjectured that when producting with a sphere the
category of the space should increase by one. Around 10 years ago Iwase
constructed a series of counterexamples, all of which had dimension at
least 10.
In this talk we construct a counterexample of dimension 7 and show that
there are no 6 dimensional counterexamples. Other counterexamples arise
from the instability of a Hopf invariant, however this example arises
since a Hopf invariant becomes stably more divisible. |