Alphabeta Math
TheoremStatement: 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.

Unoriented Thom cohomology away from two and its strict endpoint

Statement

Assume AC. Let R be a nonzero commutative unital ring with 2 invertible, r>0, MO(r)=Th(γ_r), and MSO(r)=Th(γ_r⁺). Then reduced H^i(MO(r);R)=H^{i−r}(BO(r);O_R(γ_r)). For odd r this is zero in every degree. For r=2m it is the shifted module u e R[p₁,…,p_m], with |u|=r, |e|=r and |p_i|=4i, interpreted through the oriented cover, not as a global unoriented R-Thom class. In particular reduced H^i(MO(r);R)=0 for i<2r, whereas reduced H^{2r}(MO(2m);R)=R. For MSO(r), reduced H^i(MSO(r);R)=H^{i−r}(BSO(r);R).

Facts & Assumptions

Given: AC; a nonzero commutative ring R with 2 invertible; a rank r≥1; the universal real bundle γr with its disk and sphere bundles and sign local system OR(γr); and the oriented double cover BSO(r)→BO(r) with its two-lift CW model.

[F1]

The general relative Thom isomorphism identifies reduced Thom cohomology with the cohomology of the base with coefficients in the orientation local system (General Thom isomorphism from the relative Serre spectral sequence, R-oriented vector bundle and orientation local system), and relative cohomology of the disk/sphere quotient is the reduced cohomology of the Thom space (The Thom quotient identifies relative and reduced cohomology, Thom isomorphism for oriented vector bundles).

[F2]

The finite-cover transfer identifies the orientation-local-system cohomology with the anti-invariant part of the double cover (Finite-cover transfer with inverted degree and sign anti-invariants), and the away-from-two computation gives the polynomial presentations of BO and BSO with their orientation behaviour (Cohomology of BO and BSO away from two).

[F3]

The classical tautological and oriented models supply the bundles and cells used (Stiefel spaces, Grassmannians, and tautological bundles, Oriented Grassmannians and the tautological oriented bundle), and AC is inherited from the transfer and CW model choices (The Axiom of Choice).

Proof

technique · direct
1.1givenF1F2

The finite-cover transfer identifies the right side with the anti-invariants of BSO(r) cohomology. In odd rank the polynomial generators are all orientation independent, so the anti-invariant part is zero: x=−x implies x=0 because 2 is invertible. In even rank r=2m, the polynomial presentation has anti-invariant part eR[p₁,…,p_{m-1},e²]=eR[p₁,…,p_m]. This proves the module formulas stated at the beginning of the theorem and their exact endpoint. Upstairs, the oriented Thom class changes sign under the deck map, so multiplying it by this anti-invariant Euler factor gives the invariant class u e. This explains why it descends, and why there is no unqualified unoriented R-Thom class.

2.1step 1.1F1F2F3∎

For MSO(r), the oriented Thom theorem instead gives H~^i(MSO(r);R)=H^{i-r}(BSO(r);R).

Depends on

Used by

Dependency tree · two levels

61 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