LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Oriented simplex comparison for an ordinary homology theory

Statement

For finite simplicial pairs $(K, L)$ and any ordinary theory $h$ with coefficient group $G$ , ordered simplex classes identify $C_{*}^{h} (K, L) ≅ C_{*}^{s i m p} (K, L; Z) \otimes G,$ with the alternating face differential. Consequently they give a coefficient-normalized isomorphism $h_{n} (∣ K ∣, ∣ L ∣) \to H_{n} (∣ K ∣, ∣ L ∣; G)$ , natural for simplicial maps and compatible with pair boundaries. No flatness of $G$ is assumed.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For every finite-dimensional CW pair $(X, A)$ and ordinary theory $h$ , there is a canonical isomorphism $h_{n} (X, A) ≅ H_{n} (C_{*}^{h} (X, A))$ for every integer $n$ , natural for cellular maps. It is the skeletal lift isomorphism described below and commutes with the homology connecting maps of pairs. The number of cells need not be finite. (Finite dimensional skeletal exactness computes axiomatic homology)

[F2]

For an abstract simplicial complex $K$ and an integer $n \geq 0$ , the simplicial chain group $C_{n} (K)$ is the free abelian group generated by the oriented $n$ -simplices of $K$ , subject to the relation $[v_{π (0)}, \dots, v_{π (n)}] = sgn (π) [v_{0}, \dots, v_{n}]$ for every permutation $π$ of the vertices of a simplex. For $n < 0$ , set $C_{n} (K) = 0$ . The boundary operator $\partial_{n} : C_{n} (K) \to C_{n - 1} (K)$ is $\partial_{0} = 0$ in degree $0$ . For $n \geq 1$ , it is defined on an oriented simplex by $\partial_{n} [v_{0}, \dots, v_{n}] = \sum_{i = 0}^{n} (- 1)^{i} [v_{0}, \dots, \hat{v_{i}}, \dots, v_{n}] .$ The well-definedness of this formula with respect to the chosen oriented representative is recorded in lem-simplicial-boundary-is-independent-of-oriented-representative through justified_by. (Simplicial chain groups and the boundary operator)

[F3]

If $π \in S_{n + 1}$ is odd, then $[v_{π (0)}, \dots, v_{π (n)}] = - [v_{0}, \dots, v_{n}]$ as oriented simplices. (An odd permutation reverses the sign of an oriented simplex)

[F4]

For every simplicial complex $K$ , the natural simplicial-to-singular chain map induces $H_{n}^{s i m p} (K; G) ≅ H_{n} (∣ K ∣; G)$ for all $n$ . (Simplicial and singular homology agree)

Proof

1.1

A vertex has its prescribed coefficient map $G \to h_{0} (*)$ . Inductively orient an ordered simplex $σ = [v_{0}, \dots, v_{r}]$ by the relative class whose boundary is the alternating oriented boundary of $σ$ . To justify the induction without assuming integer action, let $T$ be the union of all its faces except the face opposite $v_{0}$ . It is a cone on the boundary of that opposite face and contracts. The exact sequence of $(\partial σ, T)$ identifies ${\tilde{h}}_{r - 1} (\partial σ)$ with $h_{r - 1} (\partial σ, T)$ , which by excision is the relative group of that opposite face. This specifies uniquely the boundary class from the already oriented face and its coefficient $g$ . The cone pair boundary then specifies uniquely the relative $r$ -class. For $r = 1$ this is $(- g, g)$ at the two endpoints.

F1given

2.1

The coefficient of the opposite face is $g$ . At every common codimension-two face the boundary of this boundary is zero by the triple sequence; thus the two incident face coefficients must cancel with their inductively fixed signs. The adjacency graph of the faces of a simplex is connected, so this forces all coefficients to be $(- 1)^{j} g$ on the face omitting $v_{j}$ . For $r = 1$ the augmentation-kernel calculation supplies the same cancellation. Thus the induced differential is exactly the alternating formula of F2.

F1F2step 1.1

3.1

Permuting the vertices sends the alternating boundary class to its permutation sign times the old class, by induction on faces; injectivity of the relative-simplex boundary fixes the same sign upstairs. Adjacent transpositions generate all permutations, matching the oriented relation F3. A simplicial map injective on the vertices of a simplex hence acts by this signed ordered-image map. If its image has lower dimension, it factors through that lower-dimensional simplex and its relative degree- $r$ homology vanishes by F1. This proves simplicial naturality, including degenerate simplicial images.

F1F3step 1.1step 2.1

4.1

Summing over the simplices outside $L$ yields the asserted chain isomorphism, with no tensoring of an exact sequence required: both sides are direct sums of copies of $G$ and the differential is already the same signed matrix. Apply the skeletal homology computation F1. The simplicial-to-singular comparison F4 extends to a finite pair by the natural short exact sequences of simplicial and singular chains and the resulting pair exact sequences: the absolute isomorphisms for $K, L$ give the relative isomorphism by the usual injectivity/surjectivity exact-sequence chase. This yields the desired comparison, commuting with boundaries and normalized on a vertex. Empty complexes, $K = L$ , and dimension zero all give the same direct-sum formulas.

F1F4step 2.1step 3.1∎

Depends on

Used by

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Sources

May, A Concise Course in Algebraic Topology, 15§2, both cellular comparison theorems, pp.119–120 (standard reference, not scraped)
Hatcher, Algebraic Topology, §2C, Theorem 2C.1 setting, pp.177–179 (standard reference, not scraped)