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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The signature and the L-genus agree on products of complex projective spaces

Statement

Assume AC, inherited from the signature-product and L-genus suppliers. Let $r \geq 1$ and $k_{1}, \dots, k_{r} \geq 1$ with $k_{1} + \dots + k_{r} = k$ and let $P = {C P}^{2 k_{1}} \times \dots \times {C P}^{2 k_{r}}$ carry the product orientation. Then $σ (P) = 1 = L [P] .$

Facts & Assumptions

Given: AC, integers $k_{1}, \dots, k_{r} \geq 1$ with sum $k$ , and the product $P = {C P}^{2 k_{1}} \times \dots \times {C P}^{2 k_{r}}$ with the product orientation and product smooth structure.

[F1]

The signature is multiplicative under Cartesian products: $σ (X \times Y) = σ (X) σ (Y)$ for closed oriented manifolds of dimensions divisible by four (The signature is multiplicative under Cartesian products).

[F2]

The L-genus is a unital $Q$ -algebra homomorphism on rational oriented bordism, in particular multiplicative: $L [X \times Y] = L [X] L [Y]$ , with the product orientation (The L-genus is an oriented rational bordism ring homomorphism).

[F3]

$σ ({C P}^{2 k_{i}}) = 1 = L [{C P}^{2 k_{i}}]$ for each factor (The signature and the L-genus agree on complex projective space, The L-genus of complex projective space of even complex dimension is one).

[F4]

The product orientation and product smooth structure are those of Product orientations and Products of smooth manifolds have a canonical product smooth structure.

Proof

technique · direct; iterate multiplicativity over the factors

1.1givenF1F3F4

Signature: by [F1], applied inductively to the product and using [F4], $σ (P) = \prod_{i = 1}^{r} σ ({C P}^{2 k_{i}}) = \prod_{i = 1}^{r} 1 = 1$ by [F3].

1.2givenF2F3F4

L-genus: by [F2], $L [P] = \prod_{i = 1}^{r} L [{C P}^{2 k_{i}}] = \prod_{i = 1}^{r} 1 = 1$ by [F3].

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 give $σ (P) = 1 = L [P]$ , for every $r \geq 1$ , every partition $k_{1} + \dots + k_{r} = k$ and every $k \geq 1$ ; the products are closed oriented smooth of dimension $4 k$ and the case of a single factor is [F3].

Depends on

Used by

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Sources

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
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)