Lusternik-Schnirelmann Category

Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

“…The covering definition has been used to prove that cat.X / Ä dim.X / (see Cornea et al [5,Theorem 1.7]), but we have not seen the covering definition yield the better bound cat.X / Ä dim.X /=.conn.X /C1/. Furthermore, the introduction of Q-connectivity is new, and leads to lower upper bounds.…”

“…It is a standard consequence of the homotopy lifting property (see [5,Corollary 1.45 and Remark 1.46]) that cat. x X / Ä cat.X / for any covering x X !…”

“…LS category is an important numerical invariant in algebraic topology, critical point theory and symplectic geometry (see, for instance, Cornea et al [5], Clapp and Puppe [4] and Rudyak and Oprea [22]). One of the most basic estimates for LS category is (by Grossman [12] or see, for instance, Cornea et al [5]) (}) cat.X / Ä dim.X / c C 1 ;…”

Complexity of the forward kinematic map

“…Their work was widely extended both in analysis, most notably by J. Schwartz [23] and R. Palais [21], and in topology, by R. Fox [10], T. Ganea [11], I. James [18] and many others. Today Lusternik-Schnirelmann category is a well-developed theory with many ramifications and methods of computation techniques that allow to systematically determine the category for most of the spaces that will appear in this paper -see [4]. It is interesting to see how this very classical and independently developed theory found its application in the study of problems in robotics.…”

Lusternik–Schnirelmann category and the connectivity ofX

“…C is the LusternikSchnirelmann (LS) category. There is a wide variety of research in this area; see Cornea, Lupton, Oprea and Tanré [3], Oprea and Strom [6] and Stanley and Rodríguez Ordóñez [11]. Let X n be a space with the homotopy type of the n-skeleton of X and define cat.X n / to be the category of X n in X (see Definition 2 and Proposition 3.)…”