arXiv subjects
Publications and source records attributed to J. A. Vilches.
We develop Morse-Bott theory on posets, generalizing both discrete Morse-Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik-Schnirelmann theorem for general matchings on posets, in particular, for Morse-Bott functions.
We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the Morse-Pitcher inequalities in the context of finite spaces.
We undertake a systematic study of the notion of fibration in the setting of abstract simplicial complexes, where the concept of `homotopy' has been replaced by that of `contiguity'. Then a fibration will be a simplicial map satisfying the `contiguity lifting property'. This definition turns out to be equivalent to a known notion introduced by G. Minian, established in terms of a cylinder construction $K \times I_m$. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration have the same strong homotopy type, a notion that has been recently introduced by Barmak and Minian; any fibration with a strongly collapsible base is fibrewise trivial; and some other ones. We introduce the concept of `simplicial finite-fibration', that is, a map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration $PK \to K\times K$ is a finite-fibration, where PK is the space of Moore paths introduced by M. Grandis. This important result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we introduce a definition of `Švarc genus' of a simplicial map and, and using the properties stated before, we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers. Finally, another key result is a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber.
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.
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 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 category are obtained in the setting of finite topological spaces.