🔢Elementary Algebraic Topology Unit 7 – Simplicial Complexes

Definition and Basics

• Simplicial complexes are combinatorial objects used to represent topological spaces
• Consist of vertices (0-simplices), edges (1-simplices), triangles (2-simplices), tetrahedra (3-simplices), and higher-dimensional simplices
• Simplices are glued together along their faces to form the complex
• Must satisfy the property that any face of a simplex in the complex is also in the complex
• Provides a discrete way to study continuous spaces
  • Allows for computational methods and algorithms
  • Useful in fields like topology, geometry, and data analysis
• Can be represented abstractly as a set system or geometrically as a collection of points, lines, triangles, etc. in Euclidean space
• Subcomplex is a subset of a simplicial complex that is itself a simplicial complex

Geometric Realization

• Process of constructing a topological space from a simplicial complex
• Each abstract simplex is assigned a geometric simplex in Euclidean space
  • 0-simplex is a point, 1-simplex is a line segment, 2-simplex is a triangle, etc.
• Geometric simplices are glued together according to the face relations in the abstract complex
• Resulting space is called the geometric realization or polyhedron of the simplicial complex
• Provides a way to visualize and study the topology of the complex
• Allows for the application of continuous methods and theorems to the discrete structure
• Homeomorphic simplicial complexes have homeomorphic geometric realizations

Simplicial Maps

• Functions between simplicial complexes that preserve the simplicial structure
• Map vertices to vertices and simplices to simplices (or to lower-dimensional simplices)
• Induce continuous maps between the geometric realizations of the complexes
• Simplicial isomorphism is a bijective simplicial map whose inverse is also simplicial
  • Implies the complexes have the same combinatorial structure
• Simplicial homeomorphism is a simplicial map that induces a homeomorphism between the geometric realizations
• Useful for studying relationships and transformations between simplicial complexes
• Composition of simplicial maps is again a simplicial map
• Simplicial maps can be represented by their action on the vertices of the domain complex

Barycentric Subdivision

• Operation that refines a simplicial complex into a new complex with smaller simplices
• Barycenter of a simplex is the average of its vertices
  • Divides the simplex into smaller simplices by connecting the barycenters
• Barycentric subdivision of a complex is obtained by subdividing each simplex and gluing the resulting pieces
• Results in a homeomorphic complex with a finer triangulation
• Useful for approximating continuous maps by simplicial maps
• Can be iterated to obtain arbitrarily fine subdivisions
• Preserves the topology of the original complex
• Barycentric subdivision of a simplicial map is again a simplicial map

Simplicial Homology

• Algebraic method for studying the topological properties of simplicial complexes
• Assigns a sequence of abelian groups (homology groups) to a complex, one for each dimension
• Captures information about the "holes" in the complex
  • 0-dimensional homology measures connected components
  • 1-dimensional homology measures loops or tunnels
  • 2-dimensional homology measures voids or cavities
• Defined using the boundary maps between chain groups (formal sums of simplices)
• Homology groups are the quotients of the kernel of the boundary map by the image of the previous boundary map
• Betti numbers are the ranks of the homology groups and provide numerical invariants of the complex
• Simplicial maps induce homomorphisms between homology groups
• Homotopy equivalent complexes have isomorphic homology groups
• Can be computed algorithmically using linear algebra techniques

Applications in Topology

• Simplicial complexes are used to model and study a wide variety of topological spaces
• Triangulations of manifolds
  • Every smooth manifold admits a triangulation as a simplicial complex
  • Allows for the application of combinatorial and algebraic methods to manifold topology
• Simplicial approximation theorem
  • Every continuous map between the geometric realizations of simplicial complexes can be approximated by a simplicial map after sufficient subdivision
  • Useful for studying homotopy and homeomorphism properties
• Nerve complexes
  • Associated to a cover of a topological space, captures the intersection pattern of the cover sets
  • Nerve theorem relates the homotopy type of the space to that of the nerve complex under certain conditions
• Persistent homology
  • Studies the evolution of homology groups as a simplicial complex is built up by adding simplices
  • Useful for analyzing data sets and extracting topological features at different scales

Computational Aspects

• Simplicial complexes and their associated structures can be represented and manipulated using algorithms and data structures
• Boundary matrices encode the boundary maps between chain groups and allow for the computation of homology
• Simplicial homology can be computed using matrix reduction algorithms (Smith normal form)
• Persistent homology can be computed using the persistence algorithm, which tracks the birth and death of homology classes
• Simplicial maps can be represented by their action on vertices and implemented using data structures like hash tables or matrices
• Computational topology software (e.g., GAP, PHAT, Dionysus) provides tools for working with simplicial complexes and computing topological invariants
• Efficient data structures (e.g., simplex trees, filtrations) enable the storage and manipulation of large simplicial complexes
• Randomized and approximate algorithms can be used for handling high-dimensional and large-scale complexes

Advanced Concepts and Extensions

• Simplicial sets
  • Generalization of simplicial complexes that allows for multiple simplices with the same vertex set
  • Useful for modeling spaces with higher-dimensional symmetries or homotopy-theoretic structures
• Homology with coefficients
  • Homology groups can be computed with coefficients in any abelian group, providing additional algebraic information
  • Choice of coefficients can detect torsion and orientation properties
• Cohomology
  • Dual theory to homology, assigns abelian groups to a complex based on cochains (maps from simplices to a coefficient group)
  • Provides contravariant functoriality and cup product structure
  • Related to homology via the Universal Coefficient Theorem
• Simplicial homotopy theory
  • Studies homotopy properties of simplicial sets and their relationship to topological spaces
  • Kan complexes model spaces with well-behaved homotopy theory
  • Homotopy groups and fibrations can be studied in the simplicial setting
• Morse theory on simplicial complexes
  • Discrete analogue of smooth Morse theory, studies the relationship between the critical simplices of a function and the topology of the complex
  • Useful for computing homology and understanding the structure of the complex
• Topological data analysis
  • Applies simplicial complex techniques to the study of high-dimensional and noisy data sets
  • Persistent homology captures multiscale topological features
  • Mapper algorithm constructs simplicial models of data based on a filter function and clustering