LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The universal mod-two class detects admissible composites in the strict range

Statement

Assume AC. For $q \geq 1$ let $K_{q} = K (F_{2}, q)$ be a based CW model with its normalized fundamental class $ι_{q}$ . For $0 \leq i < q$ , the map

$η_{q, i} : A^{i} ⟶ {\tilde{H}}^{q + i} (K_{q}; F_{2}), a ⟼ a (ι_{q})$

is injective. If $f : X \to K_{q}$ classifies $z \in {\tilde{H}}^{q} (X; F_{2})$ , then $f^{*} η_{q, i} (a) = a (z)$ .

Facts & Assumptions

Given: AC; $q \geq 1$ ; a based CW model $K_{q} = K (F_{2}, q)$ with normalized fundamental class $ι_{q}$ ; an admissible-basis expansion of $a \in A^{i}$ ; and the test space $X = ({R P}^{i + 1})^{q}$ with $P = x_{1} \dots x_{q}$ .

[F1]

Eilenberg–Mac Lane representability supplies $ι_{q}$ , the classifying map of any class, and naturality of evaluation (Eilenberg--Mac Lane spaces represent singular cohomology); squares are natural (Steenrod squares are well-defined and natural).

[F2]

Every admissible sequence of degree $i$ has excess at most $i < q$ , and the leading-monomial lemma makes $a (P) \neq 0$ for nonzero $a$ (Admissible square actions have distinct leading monomials); finite products of projective spaces are finite well-pointed CW complexes.

[F3]

AC is used for the algebraic and model choices (The Axiom of Choice).

Proof

technique · direct

1.1givenF1

The published representability theorem supplies $ι_{q}$ and the classifying map, and square naturality gives the evaluation identity, also for finite linear combinations and composites. This is precisely the already-published universal-operation evaluation mechanism.

2.1step 1.1F2F3∎

Let a nonzero $a \in A^{i}$ be expanded in the admissible basis. Take $r = q$ , $L = i + 1$ , $X = ({R P}^{L})^{q}$ , and $P = x_{1} \dots x_{q}$ . This is a finite well-pointed CW complex and $P$ has degree $q$ . Every sequence appearing in $a$ has excess at most $i < q = r$ , so the leading-monomial lemma and its same-largest-term argument give $a (P) \neq 0$ . A based classifying map $f : X \to K_{q}$ exists. If $η_{q, i} (a) = 0$ , naturality would give $0 = f^{*} η_{q, i} (a) = a (P)$ , a contradiction. For $i = 0$ , the same argument is the nonzero fundamental class test. For $q = 1$ the asserted strict range contains only $i = 0$ .

Depends on

Used by

Metastable cohomology of mod-two Eilenberg–Mac Lane spaces Lemma

Dependency tree · two levels

33 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

Tom Weston, An Introduction to Cobordism Theory (standard reference, not scraped)
Allen Hatcher, Algebraic Topology (standard reference, not scraped)