Math 2.0

Algebraic topology Definition

Simplicial homology

Builds on Simplicial complexFree abelian groupKernel of a homomorphism

Definition

Let KK be a simplicial complex with a total order on its vertex set. For n≥0n \ge 0 let Cn(K)C_n(K) be the free abelian group with basis the nn-simplices of KK, and let Cn(K)=0C_n(K) = 0 for n<0n < 0. An nn-simplex with vertices v0<v1<⋯<vnv_0 < v_1 < \cdots < v_n is written [v0,…,vn][v_0, \dots, v_n]. The boundary map ∂n ⁣:Cn(K)→Cn−1(K)\partial_n \colon C_n(K) \to C_{n-1}(K) is the homomorphism given on basis elements by ∂n[v0,…,vn]=∑i=0n(−1)i[v0,…,vi^,…,vn],\partial_n [v_0, \dots, v_n] = \sum_{i=0}^{n} (-1)^i [v_0, \dots, \widehat{v_i}, \dots, v_n], where vi^\widehat{v_i} means that viv_i is omitted, and ∂0=0\partial_0 = 0. Then ∂n∘∂n+1=0\partial_{n} \circ \partial_{n+1} = 0, so (C∙(K),∂)(C_\bullet(K), \partial) is a chain complex. The nn-th simplicial homology group of KK is Hn(K)=ker⁡∂n/im⁡∂n+1.H_n(K) = \ker \partial_n / \operatorname{im} \partial_{n+1}.

Remarks

Elements of ker⁡∂n\ker \partial_n are nn-cycles and elements of im⁡∂n+1\operatorname{im} \partial_{n+1} are nn-boundaries; the kernel contains the image because ∂n∘∂n+1=0\partial_n \circ \partial_{n+1} = 0. The groups do not depend on the vertex order up to isomorphism. With coefficients in an abelian group AA, one uses Cn(K)⊗AC_n(K) \otimes A.

Example: let KK be the boundary of a triangle, with vertices 0<1<20 < 1 < 2 and edges [0,1][0,1], [0,2][0,2], [1,2][1,2]. Then ∂1[0,1]=[1]−[0]\partial_1 [0,1] = [1] - [0], ∂1[0,2]=[2]−[0]\partial_1 [0,2] = [2] - [0], ∂1[1,2]=[2]−[1]\partial_1 [1,2] = [2] - [1]. The kernel of ∂1\partial_1 is generated by [0,1]−[0,2]+[1,2][0,1] - [0,2] + [1,2], and there are no 22-simplices, so H1(K)≅ℤH_1(K) \cong \Z; also H0(K)≅ℤH_0(K) \cong \Z.

Non-example: dropping the signs (−1)i(-1)^i does not give a chain complex over ℤ\Z. For the 22-simplex, the unsigned boundary of [0,1,2][0,1,2] is [1,2]+[0,2]+[0,1][1,2] + [0,2] + [0,1], whose unsigned boundary is 2([0]+[1]+[2])≠02([0] + [1] + [2]) \ne 0.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Free abelian group, concept, no evidence levelFree abelian groupconceptGroup, concept, no evidence levelGroupconceptSimplicial complex, concept, no evidence levelSimplicial complexconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionSimplicial homology, definition, no evidence levelSimplicial homologydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Free abelian group concept
  2. Group concept
  3. Simplicial complex concept
  4. Group homomorphism definition
  5. Kernel of a homomorphism definition
  6. Simplicial homology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Simplicial_homology
About this page
TagR1RQ3226WV
URIm2://R1RQ3226WV
Kinddefinition
WritingW0 stub
MSC55N10, 55U10
DefinesSimplicial homology
UsesSimplicial complex, Free abelian group, Kernel of a homomorphism
Changed2 November 2026

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