LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Natural singular-cohomology identities are detected on finite regular complexes

Statement

Assume AC. Fix a prime $p$ and $m, n \geq 0$ . Suppose that for every space $X$ there is a map

$R_{X} : H^{m} (X; F_{p}) ⟶ H^{n} (X; F_{p})$

natural in the sense that $f^{*} R_{X} (x) = R_{K} (f^{*} x)$ for every continuous $f : K \to X$ . If $R_{K} = 0$ for every finite regular cell complex $K$ , then $R_{X} = 0$ for every space $X$ . Neither additivity of $R$ nor a simultaneous finite model for all classes is required.

Facts & Assumptions

Given: AC, the prime $p$ , nonnegative degrees $m, n$ , and the natural family $R$ in the statement.

[F1]

Singular chain groups are made of finite formal sums (Singular simplices and singular chain groups with coefficients); their boundary squares to zero, and homology is cycles modulo boundaries (The singular chain complex and singular homology).

[F2]

Over $F_{p}$ , singular cochains are the full linear dual of the singular chains (Singular cochain complex with coefficients).

[F3]

The singular-cochain coboundary is $δ ψ = ψ \partial$ (Singular cochain complex with coefficients).

[F4]

A continuous map pulls a cohomology class back by precomposition with its induced singular chain map (Singular cohomology is contravariantly functorial).

[F5]

The mod- $p$ Kronecker pairing is well-defined and natural: $⟨ f^{*} α, z ⟩ = ⟨ α, f_{*} z ⟩$ (The kronecker pairing is independent of cocycle and cycle representatives).

[F6]

AC supplies a choice function for every family of nonempty sets (The Axiom of Choice).

Proof

Proof technique: realize each individual singular cycle on a finite Delta complex, subdivide it to a finite regular complex, and use evaluation to detect the cohomology class.

1.1

Realize a mod- $p$ singular cycle on a finite regular complex. [F1] Let $z \in Z_{n} (X; F_{p})$ and write its finite support as $z = \sum_{r = 1}^{N} a_{r} σ_{r}$ . Form the finite Delta complex $P_{z}$ generated by these labeled top simplices and all their iterated face restrictions: two face occurrences are attached to the same lower simplex exactly when they are the same singular simplex of $X$ , and all attaching maps are the corresponding order-preserving affine face maps. The simplicial identities make these attachments compatible in lower dimensions. Mapping the cell labeled by a singular simplex $τ$ by $τ$ itself gives a continuous map $g : P_{z} \to X$ .

Put $ξ = \sum_{r} a_{r} [σ_{r}]_{P_{z}}$ in the Delta-chain group. For every labeled $(n - 1)$ -simplex $τ$ , its coefficient in $\partial ξ$ is exactly the coefficient of the singular basis element $τ$ in $\partial z$ , hence is zero in $F_{p}$ . Thus $ξ$ is a mod- $p$ cycle and $g_{*} [ξ] = [z]$ . This construction also covers $n = 0$ : $P_{z}$ is the finite discrete set of labeled vertices in the support, carrying their coefficients $a_{r}$ .

The second barycentric subdivision $K_{z}$ of a Delta complex is a finite simplicial complex, hence a finite regular cell complex. The affine subdivision operator is a chain map and the cone calculation $\partial T + T \partial = 1 - S$ makes it chain-homotopic to the identity. Therefore the subdivided cycle $ζ$ and the composite $f : K_{z} \to P_{z} \to X$ satisfy

$f_{*} [ζ] = [z] .$

All face identifications, coefficient operations, and subdivisions here are finite prescribed operations; no choice principle is used.

1.2

Prove that evaluation detects a mod- $p$ cohomology class. [F2, F3, F6] Let a degree- $n$ cocycle $φ$ vanish on every degree- $n$ cycle. If $n = 0$ , every zero-chain is a cycle and hence $φ = 0$ . Suppose $n > 0$ . For $u \in B_{n - 1} = \partial C_{n} (X; F_{p})$ , choose any $c$ with $\partial c = u$ and define $b_{0} (u) = φ (c)$ . This is well-defined: two choices differ by a cycle, on which $φ$ vanishes. It is linear by using sums and scalar multiples of preimages. By [F6], choose a vector-space complement $M$ with $C_{n - 1} = B_{n - 1} \oplus M$ , and extend $b_{0}$ by zero on $M$ to a cochain $b$ . Then [F3] gives

$δ b (c) = b (\partial c) = b_{0} (\partial c) = φ (c)$

for every $c \in C_{n}$ . Thus $φ = δ b$ . Consequently, if a class in $H^{n} (X; F_{p})$ pairs to zero with every homology class, it is zero. The sole AC use is the complement of the possibly infinite-dimensional boundary subspace.

2.1

Apply the finite hypothesis to every evaluation cycle. [given, F4, F5, step 1.1, step 1.2] Fix a space $X$ and $x \in H^{m} (X; F_{p})$ , and put $α = R_{X} (x)$ . For any mod- $p$ $n$ -cycle $z$ , choose $f : K_{z} \to X$ and $ζ$ as in step 1.1. Naturality of $R$ and the assumed finite-complex vanishing give

$f^{*} α = f^{*} R_{X} (x) = R_{K_{z}} (f^{*} x) = 0.$

By [F5] and $f_{*} [ζ] = [z]$ ,

$⟨ α, [z] ⟩ = ⟨ α, f_{*} [ζ] ⟩ = ⟨ f^{*} α, [ζ] ⟩ = 0.$

Step 1.2 now yields $α = 0$ . Since $X$ and $x$ were arbitrary, $R_{X} = 0$ for every space.

3.1

Check empty, zero, endpoint, and degeneracy cases. [F1, F2, F3, F6, step 1.1, step 1.2, step 2.1] If $X$ is empty, its chain, homology, and cohomology groups in the stated degrees are zero. The zero cycle may be represented by the empty finite complex and evaluates to zero. The case $n = 0$ was handled separately in the evaluation-detection step; degree $m = 0$ requires no change because naturality alone is used on the input. A one-term zero-cycle with any nonzero coefficient is represented by one weighted vertex. Degenerate singular simplices are still finite basis elements and may label cells whose map to $X$ is degenerate. Both implications in the displayed naturality equality are literal equalities, not directions of a biconditional. Finite face identification and subdivision use no choice; AC is used only for the complement in step 1.2. ∎

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
N. E. Steenrod and D. B. A. Epstein, Cohomology Operations (standard reference, not scraped)