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

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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.

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The signature and the L-genus agree on complex projective space

Statement

Assume AC, inherited from the signature form and L-genus suppliers. For every $k \geq 0$ , with the complex orientation of ${C P}^{2 k}$ , $σ ({C P}^{2 k}) = 1 = L [{C P}^{2 k}] .$

Facts & Assumptions

Given: AC, the integer $k \geq 0$ , the complex orientation of ${C P}^{2 k}$ , and the generator $y = c_{1} (γ^{*})$ .

[F1]

$σ (M) = p - q$ is the difference of the positive and negative inertia indices of the nondegenerate symmetric form $Q_{M}$ on the middle cohomology (The signature of a closed oriented manifold of dimension divisible by four, The middle-dimensional intersection form is symmetric and nondegenerate).

[F2]

$Q_{M} (x, y) = ⟨ x ⌣ y, [M] ⟩$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, Kronecker evaluation pairing).

[F3]

The in-run supplier gives $H^{*} ({C P}^{n}; Z) = Z [y] / (y^{n + 1})$ , and the coefficient bridge of The signature is independent of the diagonalizing basis and unchanged by scalar extension from the rationals to the reals gives $H^{2 k} ({C P}^{2 k}; R) = R y^{k}$ , and $⟨ y^{2 k}, [{C P}^{2 k}] ⟩ = 1$ (The tangent bundle of complex projective space and its Pontryagin classes).

[F4]

The L-genus satisfies $L [{C P}^{2 k}] = 1$ for every $k \geq 0$ (The L-genus of complex projective space of even complex dimension is one).

Proof

technique · direct; the middle cohomology is one-dimensional with value one

1.1givenF1F2F3

By [F3], $H^{2 k} ({C P}^{2 k}; R)$ is the one-dimensional space spanned by $y^{k}$ , and $Q_{{C P}^{2 k}} (y^{k}, y^{k}) = ⟨ y^{2 k}, [{C P}^{2 k}] ⟩ = 1$ by [F2]. Hence the matrix of $Q$ in the basis ${y^{k}}$ is the $1 \times 1$ matrix $(1)$ , its inertia is $(1, 0, 0)$ , and [F1] gives $σ ({C P}^{2 k}) = 1 - 0 = 1$ .

1.2givenF4

The L-genus value is $L [{C P}^{2 k}] = 1$ by [F4].

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 give $σ ({C P}^{2 k}) = 1 = L [{C P}^{2 k}]$ for every $k \geq 0$ ; for $k = 0$ the manifold is a point, the middle group is $H^{0} = R$ with $Q (1, 1) = 1$ , and both values are $1$ .

Depends on

Used by

Dependency tree · two levels

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Sources

John Milnor and James Stasheff, Characteristic Classes (re-typeset scan; original pagination) (standard reference, not scraped)
Tom Weston, An Introduction to Cobordism Theory (lecture notes, Stanford) (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)