Ganea's problems on Lusternik-Schnirelmann category

岩瀬 則夫(九州大学数理学研究科)


Problem 2 paused by Tudor Ganea is a question on Lusternik-Schnirelmann category, or LS category for short: Is the LS category of a
space increased by 1 by taking the product with a sphere? The affirmative answer was usually supposed to be true and come to be
called as 'the Ganea conjecture'. However, under a condition between dimension and LS category, the criterion for Ganea's conjecture
on LS category is obtained, using the stabilised higher Hopf invariants. This allows us to construct a series of complexes Qp indexed by
all the primes p with cat Qp = 2 and cat QpxSn = 2 for either n > 1 or n = 1 and p = 2. This disproves Ganea's conjecture on LS category.
As an application, conditions in terms of homotopy invariants of the attaching maps are given to determine LS category of sphere-