We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

Abstract. In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and it only depends on the simplicial structure rather than its geometric realization.In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex.Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of LScategory are obtained in the setting of finite topological spaces.

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The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U (n), U (2n)/Sp(n) and U (n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of Sp (2) by categorical open sets. The obstacles to generalize these results to Sp(n) are discussed.

We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new invariant with those of simplicial and topological LS-category.