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

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The finite Thom classifying detector map exists and is continuous

Statement

Assume AC and the construction of Finite Thom classifying detector map, using the single global basis B=⋃{d≥0}B_d and section fixed there. For every r≥2, the initial segment B(r)=⋃{0≤d<r}B_d is finite; for each b∈B_d with d_b=d, the coordinate class m̄_{b,r}∈H̃^{r+d_b}(T_r;F₂) has a based representative f_{r,b}:T_r→K(F₂,r+d_b) representing that class; and the coordinate family uniquely defines a continuous based map f_r:T_r→P_r.

Facts & Assumptions

Given: AC and the construction of Finite Thom classifying detector map, using the single global basis $B = ⨆_{d \geq 0} B_{d}$ and the degree-preserving section $s$ fixed there; a rank $r \geq 2$ .

[F1]

The definition fixes the global basis and section and the initial segments $B (r)$ , and the freeness theorem makes the lifts a free $A$ -basis (Finite Thom classifying detector map, Stable unoriented Thom cohomology is free over the square algebra).

[F2]

The degreewise constancy lemma identifies each fixed $m_{b}$ with an actual reduced cohomology class of $T_{r}$ once $r > d_{b}$ , and the degree pieces of $M$ are finite-dimensional (Stable universal Thom cohomology is eventually constant in every degree, Degreewise mod-two cohomology of the universal real Thom prespectrum).

[F3]

Eilenberg–Mac Lane representability turns a reduced cohomology class on a based CW complex into a based homotopy class with an actual representative, and the finite product universal property assembles the coordinate maps into a unique continuous based map (Eilenberg--Mac Lane spaces represent singular cohomology, A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice); AC underlies the global choices (The Axiom of Choice).

Proof

technique · direct

1.1givenF1F2F3

This initial segment is finite: each Q^d is a quotient of the finite-dimensional M^d, and there are only r degrees in the union. For b∈B_d, the stable coordinate isomorphism M^d→H̃^{r+d}(T_r;F₂) supplies the rank-r class m_{r,b} from the same fixed stable element m_b. Let f_{r,b}:T_r→K(F₂,r+d) classify m_{r,b}, using the published Eilenberg–Mac Lane representability theorem. Set $P_{r} = \prod_{b \in B (r)} K (F_{2}, r + d_{b})$ and $f_{r} = (f_{r, b})_{b \in B (r)}$ .

2.1step 1.1F3∎

The product is finite, so its coordinate maps define a continuous based map. This is the promised construction from homogeneous free-module generators; the generic representability construction alone would not show the comparison.

Depends on

Used by

The finite Thom detector is a mod-two cohomology isomorphism below 2r Theorem

Cited to discharge well-definedness by Finite Thom classifying detector map.

Dependency tree · two levels

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Sources

Tom Weston, An Introduction to Cobordism Theory (standard reference, not scraped)