The geometric realization of an abstract simplicial complex

Depends on

• An abstract simplicial complex
• The geometric simplex spanned by affinely independent vertices
• The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
• Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Used by

• A simplicial map and its geometric realization Definition
• Canonical barycentric realization map Definition
• Finite convex cell complex and linear subdivision Definition
• Open and closed stars in a subdivision Definition
• Spherical subsets, the nerve, the poset of spherical cosets, and the Davis realization Definition
• The affine characteristic singular simplex of an ordered simplex Definition
• The degreewise simplicial-to-singular homomorphisms Definition
• Weak-join classifying model of a discrete group Definition
• Incidence cochains of a closed interval Example
• A finite triangulated surface has a one-polygon schema Lemma
• A geodesically convex cover has the homotopy type of its nerve Lemma
• A polyhedral dimension bound for every sheaf Lemma
• Based loop spaces of CW complexes have CW homotopy type Lemma
• Compact subsets of an arbitrary simplicial realization meet finitely many open simplices Lemma
• Face coherence, global hat coordinates and a uniform star radius Lemma
• Finite cw pairs admit finite simplicial homotopy models Lemma
• Finite simplicial constructible sheaves have an actual derived incidence model Lemma
• Finite simplicial maps have explicit compatible triangulated mapping cylinders Lemma
• Finite simplicial open stars have explicit conical product charts Lemma
• Finite simplicial weak topology agrees with euclidean topology Lemma
• Finite triangulation of a compact connected surface Lemma
• General linear frame spaces have the Stiefel cohomology gap Lemma
• Geometric simplices intersect in the realization of their common face Lemma
• Intersection of root subcomplexes and purity under convexity Lemma
• Smooth manifolds have CW homotopy type Lemma
• The factorization criterion, linear independence of the faces, and the geometric simplicial structure of X(sigma) Lemma
• A finite simplicial complex has a compact Hausdorff realization Proposition
• Barycentric subdivision realizes homeomorphically Theorem
• Simplicial and singular homology agree Theorem
• The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Allen Hatcher, Algebraic Topology (standard reference, not scraped)
• Vidit Nanda, Computational Algebraic Topology, Lecture 01: Complexes (standard reference, not scraped)