ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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A strict metastable Eilenberg–Mac Lane range

Example

Assume AC. For K=K(F₂,3), H³(K;F₂)=F₂{ι₃}, H⁴(K;F₂)=F₂{Sq¹ι₃}, and H⁵(K;F₂)=F₂{Sq²ι₃}. The strict operation range has i<3; at degree 6 the polynomial presentation also has ι₃², so the endpoint is excluded.

Facts & Assumptions

Given: AC; the model $K = K (F_{2}, 3)$ ; the strict metastable range $i < 3$ ; and the low-degree admissible basis $A^{0} = {1}$ , $A^{1} = {S q^{1}}$ , $A^{2} = {S q^{2}}$ .

[F1]

The strict metastable theorem identifies ${\tilde{H}}^{3 + i} (K; F_{2})$ with $A^{i}$ for $0 \leq i < 3$ by evaluation on $ι_{3}$ (Metastable cohomology of mod-two Eilenberg–Mac Lane spaces).

[F2]

The admissible composites form a basis of $A^{i}$ in each degree (Admissible composites present the mod-two square algebra), and the full polynomial presentation of $H^{*} (K; F_{2})$ gives the degree-six generators (Polynomial mod-two cohomology of Eilenberg–Mac Lane spaces).

Verification

1.1givenF1F2

The metastable theorem identifies each listed group with A^i by evaluation on ι₃. The low-degree admissible basis gives A⁰={1}, A¹={Sq¹}, A²={Sq²}; the normalized universal class and naturality identify the three images.

2.1step 1.1F1F2∎

The strict inequality excludes i=3, so the example does not extend the commissioned comparison range.

Depends on

Used by

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Sources

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