A triangulation of a topological space is a homeomorphism from the realization of a simplicial complex to that space. In three-manifold topology, thickening its one-skeleton and the dual one-skeleton constructs a Heegaard splitting.
An element of the underlying vertex set of an abstract simplicial complex; a simplex is a finite subset of these vertices. In a barycentric subdivision, the vertices are nonempty simplices of the original complex.
A simplicial subcomplex is a subfamily that is itself a simplicial complex. In particular, every face of a simplex in also lies in .
The geometric realization glues one geometric simplex for every abstract simplex of , identifying their common faces.
The open star of a vertex is the union of the relative interiors of the simplices containing . The open stars of the vertices cover the geometric realization.
A simplicial map sends vertices to vertices and sends the vertex set of every simplex to the vertex set of a simplex. It induces a continuous map between geometric realizations and a chain map between simplicial chain complexes.
If simplicially approximates , then and lie in a common simplex for every . The affine formulatherefore remains in the realization and gives a homotopy from to .
For a finite simplicial complex , a simplicial complex , and a continuous map , some iterated barycentric subdivision admits a simplicial approximation to . The proof applies the Lebesgue number lemma to the inverse images of target open stars and uses that the mesh of a simplicial complex tends to zero under repeated barycentric subdivision.
If , every continuous map is null-homotopic. After simplicial approximation, its image lies in the -skeleton of a triangulation of , hence omits a point. The punctured sphere is homeomorphic to and is therefore contractible.
Simplicial maps are contiguous when lies in one simplex of for every simplex of . Contiguous maps have homotopic geometric realizations.
The probability simplex consists of vectors of nonnegative weights summing to one. It is the convex hull of the coordinate vectors. A linear function has maximum on this simplex; equality holds exactly when the weights are supported on maximizing indices.
The boundary subcomplex consists of all proper faces of the -simplex. Its geometric realization of a simplicial complex is homeomorphic to the sphere , while is homeomorphic to the closed disk .
A regular simplex has pairwise equidistant vertices. Conversely, every finite equidistant subset of Euclidean space is the vertex set of a regular simplex.
A pure -dimensional simplicial complex is a pseudomanifold when every codimension-one simplex belongs to exactly two top-dimensional simplices and the top simplices are connected through shared codimension-one faces.
Compatible orientations of all top-dimensional simplices make their signed sum a cycle. For a connected pseudomanifold this cycle generates top homology over the integers; if no compatible orientation exists, top integral homology is zero.
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A simplicial complex is a mathematical structure used in algebraic topology and combinatorial mathematics to study spaces and their properties. It is a way of building up a geometric object from simpler building blocks called simplices. ### Definition of a Simplicial Complex A simplicial complex \( K \) is a set of simplices that satisfies two conditions: 1. **Non-emptiness**: The empty set is in \( K \).