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The eight-dimensional signature formula

Statement

Assume AC, inherited from the Hirzebruch signature theorem. For every closed oriented smooth $8$ -manifold $M$ , $σ (M) = \frac{1}{45} (7 p_{2} [M] - p_{(1, 1)} [M]) .$ Here $p_{(1, 1)} [M] : = ⟨ p_{1} (T M) ⌣ p_{1} (T M), [M] ⟩$ ; it is not the square of a degree-four evaluation on $[M]$ . Consequently $45$ divides $7 p_{2} [M] - p_{(1, 1)} [M]$ in $Z$ .

Facts & Assumptions

Given: AC; a closed oriented smooth $8$ -manifold $M$ , and the Pontryagin classes $p_{1}, p_{2}$ of $T M$ .

[F1]

$σ (M) = L [M] = ⟨ L_{2} (T M), [M] ⟩$ (The Hirzebruch signature theorem).

[F2]

$L_{2} = (7 p_{2} - p_{1}^{2}) / 45$ (The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

[F3]

The Pontryagin numbers $p_{(1, 1)} [M] = ⟨ p_{1} ⌣ p_{1}, [M] ⟩$ and $p_{2} [M] = ⟨ p_{2}, [M] ⟩$ are integers. The Kronecker pairing is $Q$ -linear in the cohomology variable; no multiplicativity of evaluation on a single fundamental class is asserted (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

[F4]

$σ (M)$ is an integer by The signature of a closed oriented manifold of dimension divisible by four.

Proof

technique · direct; specialise the signature theorem to $k=2$

1.1givenF1F2F3

By [F1], [F2] and the linearity of the Kronecker pairing [F3], $σ (M) = ⟨ (7 p_{2} - p_{1}^{2}) / 45, [M] ⟩ = \frac{1}{45} (7 ⟨ p_{2}, [M] ⟩ - ⟨ p_{1} ⌣ p_{1}, [M] ⟩) = \frac{1}{45} (7 p_{2} [M] - p_{(1, 1)} [M])$ .

2.1step 1.1F3F4

Divisibility: the left side is an integer by [F4] and $7 p_{2} [M] - p_{(1, 1)} [M]$ is an integer by [F3], so $45$ divides $7 p_{2} [M] - p_{(1, 1)} [M]$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 prove the displayed formula and the divisibility statement.

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)