Variation zum Konzept der Lusternik—Schnirelmann—Kategorie

For any non—empty class a of pointed spaces, we define the notions of a—cat and a—cat in analogy to T. Ganea's definitions of cat and Cat. Let e.g. s be the class of wedges of spheres Sⁿ, n ≥ 1, then the difference s—cat — cat can be arbitrarily large, s—Cat is well known as spherical cone—length, but what is s—Cat — s—cat? If σ is the class of suspensions we recover cat and Cat. We also investigate subclasses a ? σ such that a—cat(x) = σ–cat(X) for large classes of spaces X. As a byproduct we extend known mapping theorems for cat slightly. As an application we obtain a—cat as an upper bound for the solvability (or nilpotency) degree of certain groups of self—equivalences.