On Foxʼs m-dimensional category and theorems of Bochner type

Homotopy Invariants and Almost Non-Negative Curvature

“…Recall that Bochner's Theorem states that a compact manifold M with non-negative Ricci curvature obeys a Betti number condition: b 1 pM q ď dimpM q. This type of inequality was refined in [18] (also see [19]) with dimpM q being replaced by catpM q. Here we have sharper information which is a topological analogue of Yamaguchi's theorem [23].…”

Aspects of Lusternik-Schnirelmann Category

“…By the cuplength and product inequalities for category, it is easy to see that catpS 2ˆT 2 q " 3. As shown in [17], cat 1 pS 2ˆT 2 q " 2. Now, observe that, if H denotes either the northern or southern hemisphere of S 2 union a small open collar, then for the covering map p : S 2ˆR2 Ñ S 2ˆT 2 , p´1pHˆT 2 q " HˆR 2 is contractible.…”

“…The proof of this follows from the properties of an invariant called category weight derived from the reformulation diagram. See [5,17] for example. 4.…”

“…This property is, more or less, a general property of category-like invariants. For cat 1 , see [16,17].…”

“…LS category is an important numerical invariant in algebraic topology, critical point theory and symplectic geometry (see [5,12,15]). Furthermore, various "forms" of category are now finding use in areas ranging from differential geometry [17] to robotics and motion planning [9][10][11]. In this paper, which is in part a survey, we concentrate on how LS category and its offshoots interact with the fundamental group.…”