Mathematics.

homology

Simplicial Homology

Algebraic Topology65 minDifficulty7 out of 10

You should know: singular homology, chain complexes, linear independence

Overview

Simplicial homology is a computational approach to homology theory defined on triangulated spaces called simplicial complexes (or delta complexes). By decomposing a space into simplices -- points, edges, triangles, tetrahedra, and their higher-dimensional analogues -- one can define chain groups and boundary maps explicitly with integer matrices, making homology algorithmically computable. Simplicial homology agrees with singular homology for triangulable spaces.

Intuition

Break a shape into building blocks: vertices (0-simplices), edges (1-simplices), triangles (2-simplices), tetrahedra (3-simplices). Each edge has two boundary vertices; each triangle has three boundary edges; each tetrahedron has four boundary triangular faces. Summing boundaries with signs gives a 'boundary operator'. A cycle is a collection of simplices with no boundary; a boundary is a cycle that itself bounds something higher-dimensional. Homology = cycles modulo boundaries, measuring the 'holes' that are not filled in.

Formal Definition

Definition

A simplicial complex K is a collection of simplices closed under taking faces. The n-th chain group C_n(K) is the free abelian group on the oriented n-simplices. The boundary operator ∂_n: C_n → C_{n-1} sends each n-simplex [v₀,...,vₙ] to the alternating sum of its faces. The simplicial homology groups are H_n(K) = ker(∂_n) / im(∂_{n+1}).

Cn(K)=⨁σ∈K,dim⁡σ=nZ⋅σC_n(K) = \bigoplus_{\sigma \in K, \dim\sigma=n} \mathbb{Z}\cdot \sigma
Chain group
∂n[v0,…,vn]=∑i=0n(−1)i[v0,…,v^i,…,vn]\partial_n[v_0,\ldots,v_n] = \sum_{i=0}^n (-1)^i [v_0,\ldots,\hat{v}_i,\ldots,v_n]
Boundary operator
∂n−1∘∂n=0\partial_{n-1} \circ \partial_n = 0
Boundary of boundary is zero
Hn(K)=ker⁡(∂n)/im(∂n+1)H_n(K) = \ker(\partial_n) / \mathrm{im}(\partial_{n+1})
Simplicial homology

Notation

NotationMeaning
Cn(K)C_n(K)n-th simplicial chain group of complex K
∂n\partial_nBoundary homomorphism C_n → C_{n-1}
[v0,…,vn][v_0,\ldots,v_n]Oriented n-simplex with vertices v₀,...,vₙ
Hn(K)H_n(K)n-th simplicial homology group of K
βn\beta_nn-th Betti number: rank of H_n(K)

Theorems

Theorem 1: Simplicial equals Singular Homology
ForanysimplicialcomplexK,thesimplicialhomologygroupsHn(K)arenaturallyisomorphictothesingularhomologygroupsHn(∣K∣)ofthegeometricrealization∣K∣.For any simplicial complex K, the simplicial homology groups H_n(K) are naturally isomorphic to the singular homology groups H_n(|K|) of the geometric realization |K|.
Theorem 2: Euler Characteristic Formula
ForafinitesimplicialcomplexK,theEulercharacteristicχ(K)=Σn(−1)n⋅∣n−simplices∣=Σn(−1)n⋅rank(Hn(K))=Σn(−1)n⋅βn.For a finite simplicial complex K, the Euler characteristic χ(K) = Σ_n (-1)^n · |{n-simplices}| = Σ_n (-1)^n · rank(H_n(K)) = Σ_n (-1)^n · β_n.
Theorem 3: Invariance under Subdivision
IfK′isasimplicialsubdivisionofK,thenHn(K′)≅Hn(K)foralln.If K' is a simplicial subdivision of K, then H_n(K') ≅ H_n(K) for all n.

Worked Examples

  1. 1

    Label vertices v₀, v₁, v₂ and edges e₀=[v₁,v₂], e₁=[v₀,v₂], e₂=[v₀,v₁] (no 2-simplex).

  2. 2

    C₀ = Z³ with basis {v₀,v₁,v₂}, C₁ = Z³ with basis {e₀,e₁,e₂}, C₂ = 0.

  3. 3

    Boundary: ∂₁(e₀)=v₂-v₁, ∂₁(e₁)=v₂-v₀, ∂₁(e₂)=v₁-v₀.

    ∂1=(−1−1−110−1011)\partial_1 = \begin{pmatrix} -1 & -1 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & 1 \end{pmatrix}
  4. 4

    H₀ = Z (one connected component). ker(∂₁) has rank 1 (the cycle e₀-e₁+e₂). im(∂₂)=0. So H₁ = Z.

✓ Answer

H₀(S^1) = Z, H₁(S^1) = Z, H_n(S^1) = 0 for n ≥ 2.

Practice Problems

Mediumproof writing

Prove that ∂_{n-1} ∘ ∂_n = 0 for simplicial chain complexes.

Mediumapplication

A simplicial complex K has 5 vertices, 8 edges, and 4 triangles. Compute χ(K) and the Betti numbers assuming K is connected and simply connected.

Common Mistakes

Common Mistake

Orientation of simplices does not matter in simplicial homology.

Orientation is essential. Changing the orientation of a simplex negates it as a chain. However, the homology groups H_n(K) are well-defined independent of the choice of orientations.

Common Mistake

Simplicial homology computes something different from singular homology.

For triangulable spaces, simplicial and singular homology agree. Simplicial homology is computationally more tractable, while singular homology is defined for all topological spaces.

Quiz

The boundary of a 2-simplex [v₀, v₁, v₂] is:
For a triangulation of S^2 with V vertices, E edges, F triangular faces, the Euler characteristic V - E + F equals:
Which of the following is NOT a homotopy invariant?

Historical Background

Simplicial homology was the first rigorous formulation of homology, growing out of Poincaré's use of polyhedra in Analysis Situs (1895). Poincaré introduced the Betti numbers (ranks of homology groups) and the Euler characteristic. Veblen, Alexander, and Lefschetz systematized the theory in the 1920s--1930s. The comparison with singular homology was established by Eilenberg and Steenrod's axiomatic approach.

  1. 1895

    Poincaré defines Betti numbers and homology for polyhedra

    Poincaré

  2. 1923

    Veblen publishes a rigorous treatment of simplicial homology

    Veblen

  3. 1952

    Eilenberg and Steenrod prove simplicial = singular homology via axioms

    Eilenberg, Steenrod

Summary

  • Simplicial homology decomposes a triangulated space into simplices and defines chain groups, boundary operators, and homology groups H_n = ker(∂_n)/im(∂_{n+1}).
  • The boundary operator satisfies ∂∘∂ = 0, which is the key algebraic identity enabling the definition of homology.
  • Simplicial homology agrees with singular homology for triangulable spaces.
  • The Euler characteristic equals both the alternating sum of simplex counts and the alternating sum of Betti numbers.
  • Computing simplicial homology reduces to linear algebra: finding kernels and images of integer matrices.

References

  1. BookHatcher, A. Algebraic Topology. Cambridge University Press, 2002. Section 2.1.
  2. BookMunkres, J.R. Elements of Algebraic Topology. Addison-Wesley, 1984.
  3. WebsiteWikipedia: Simplicial homology