7.1 Simplices and simplicial complexes

Simplices are the building blocks of simplicial complexes, generalizing triangles and tetrahedra to higher dimensions. They're defined as convex hulls of affinely independent points, with dimensions ranging from 0 (points) to n (n-simplices).

Simplicial complexes are topological spaces constructed by gluing simplices together, following specific rules. They're crucial in algebraic topology, allowing us to represent complex structures using simple geometric shapes and study their properties through combinatorial methods.

Simplices and their dimensions

Definition and basic properties

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• Simplex generalizes triangle or tetrahedron to arbitrary dimensions defined as convex hull of (n+1) affinely independent points in Euclidean space
• Dimension of simplex equals one less than number of vertices (0-simplex point, 1-simplex line segment, 2-simplex triangle, 3-simplex tetrahedron)
• Represent k-simplex as [v0, v1, ..., vk] where vi are vertices and ordering determines orientation
• Standard n-simplex subset of R^(n+1) where sum of all coordinates equals 1 and all coordinates non-negative
• Simplices serve as building blocks for more complex topological structures in simplicial complexes

Coordinates and boundaries

• Barycentric coordinates uniquely specify any point within simplex as weighted sum of vertices
• Boundary of k-simplex consists of all (k-1)-dimensional faces forming simplicial complex
• Calculate barycentric coordinates using formula: $p = \sum_{i=0}^n \lambda_i v_i$ where $\sum_{i=0}^n \lambda_i = 1$ and $\lambda_i \geq 0$
• Boundary operator $\partial$ maps k-simplex to sum of its (k-1)-faces with alternating signs based on vertex ordering

Examples and applications

• 0-simplex [v0] represents single point
• 1-simplex [v0, v1] represents line segment between two points
• 2-simplex [v0, v1, v2] represents triangle with vertices v0, v1, v2
• 3-simplex [v0, v1, v2, v3] represents tetrahedron with vertices v0, v1, v2, v3
• Higher-dimensional simplices used in data analysis (simplicial complex models for high-dimensional data)
• Simplices form basis for simplicial homology theory in algebraic topology

Constructing simplicial complexes

Fundamental properties and construction

• Simplicial complex topological space constructed by gluing simplices along faces subject to specific rules
• Two fundamental properties:
  • Every face of simplex in complex also in complex
  • Intersection of any two simplices either empty or face of both
• Construct simplicial complex:
  1. Start with collection of simplices
  2. Ensure all faces included
  3. Verify intersections of simplices are faces of both
• Dimension of simplicial complex equals maximum dimension of constituent simplices
• Represent simplicial complex by vertex set and list of maximal simplices (not contained in larger simplex)

Representation and description

• F-vector records number of simplices of each dimension providing compact description of structure
• Calculate f-vector components: $f_i = \text{number of i-dimensional simplices}$
• Abstract simplicial complexes generalize geometric simplicial complexes defined by vertex sets and collections of subsets representing simplices
• Nerve of a cover important construction method for simplicial complexes from topological data

Examples of simplicial complexes

• Triangle with its edges and vertices forms 2-dimensional simplicial complex
• Tetrahedron with all faces, edges, and vertices forms 3-dimensional simplicial complex
• Simplicial complex modeling social network:
  • Vertices represent individuals
  • Edges represent connections
  • Triangles represent groups of three mutually connected individuals
• Vietoris-Rips complex constructed from point cloud data used in topological data analysis

Faces and cofaces of simplices

Faces and their properties

• Face of k-simplex any l-simplex (l ≤ k) formed by subset of vertices including simplex itself
• Proper faces all faces except simplex itself and empty set
• Calculate number of l-faces of k-simplex using binomial coefficient: $\binom{k+1}{l+1}$
• Boundary operator $\partial$ maps simplex to alternating sum of its faces

Cofaces and related concepts

• Coface of simplex σ in simplicial complex any simplex τ containing σ as face
• Link of simplex set of all simplices disjoint from it but together form larger simplex in complex
• Star of simplex union of all simplices having it as face forming subcomplex of original complex
• Facets maximal faces of simplicial complex not contained in larger simplex of complex

Applications and examples

• Faces and cofaces crucial for understanding combinatorial structure of simplicial complexes
• Use faces and cofaces to define various topological operations (homology groups)
• Example: 2-simplex [v0, v1, v2] has faces:
  • 0-faces: [v0], [v1], [v2]
  • 1-faces: [v0, v1], [v1, v2], [v0, v2]
  • 2-face: [v0, v1, v2] itself
• Link of vertex in triangulation of surface consists of edges and vertices forming cycle around vertex

Boundary and interior of simplicial complexes

Boundary definition and properties

• Boundary of simplicial complex consists of simplices that are faces of odd number of maximal simplices
• Closed simplicial complex has no boundary
• Calculate boundary using chain complex and boundary operator
• Boundary of n-simplex consists of all its (n-1)-dimensional faces

Interior and closure

• Interior of simplicial complex complement of boundary within complex itself
• Open simplicial complex equals its interior
• Closure of subset A of simplicial complex K smallest subcomplex of K containing A
• Interior point of simplex point not contained in any proper face

Geometric realization and topological concepts

• Geometric realization views simplicial complex as topological space
• Boundary and interior have usual topological meanings in geometric realization
• Euler characteristic key topological invariant related to boundary structure
• Calculate Euler characteristic using f-vector: $\chi = \sum_{i=0}^n (-1)^i f_i$
• Relative homology groups of simplicial complex with respect to boundary provide important topological information
• Compute relative homology groups using chain complexes of simplicial complex and its boundary