岩瀬 則夫（九州大学数理学研究科）

abstract:

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-

bundles-over-spheres: