5,218 research outputs found
Lusternik-Schnirelmann category of simplicial complexes and finite spaces
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.Comment: 18 pages, 10 figures, this is a new version with some minor changes
and a new exampl
An elementary illustrated introduction to simplicial sets
This is an expository introduction to simplicial sets and simplicial homotopy
theory with particular focus on relating the combinatorial aspects of the
theory to their geometric/topological origins. It is intended to be accessible
to students familiar with just the fundamentals of algebraic topology.Comment: 57 pages, 32 figures. Further corrections and additions. Section 2
has been reorganized with new material added. Section 5.1 on Simplicial Hom
added. Hopefully final version. 10/3/16-Added errata section at end and made
minor correction
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