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The four-dimensional signature formula

Statement

Assume AC, inherited from the Hirzebruch signature theorem. For every closed oriented smooth $4$ -manifold $M$ , $σ (M) = \frac{1}{3} ⟨ p_{1} (T M), [M] ⟩ = \frac{p_{1} [M]}{3} .$ Consequently $3$ divides the Pontryagin number $p_{1} [M] \in Z$ .

Facts & Assumptions

Given: AC; a closed oriented smooth $4$ -manifold $M$ .

[F1]

$σ (M) = L [M] = ⟨ L_{1} (T M), [M] ⟩$ (The Hirzebruch signature theorem).

[F2]

$L_{1} = p_{1} / 3$ (The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

[F3]

$p_{1} [M] = ⟨ p_{1} (T M), [M] ⟩ \in Z$ is the first Pontryagin number, computed by the Kronecker pairing, which is linear (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

[F4]

The signature $σ (M) = p - q$ is by definition the difference of two nonnegative integers, hence an integer (The signature of a closed oriented manifold of dimension divisible by four).

Proof

technique · direct; specialise the signature theorem to $k=1$

1.1givenF1F2F3

By [F1] and [F2], $σ (M) = ⟨ L_{1} (T M), [M] ⟩ = ⟨ p_{1} (T M) / 3, [M] ⟩ = \frac{1}{3} ⟨ p_{1} (T M), [M] ⟩ = p_{1} [M] / 3$ , using the $Q$ -linearity of the Kronecker pairing [F3].

2.1step 1.1F3F4

Divisibility: $σ (M) \in Z$ and $p_{1} [M] \in Z$ with $3 σ (M) = p_{1} [M]$ , so $3$ divides $p_{1} [M]$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 prove the displayed formula and the divisibility statement for every closed oriented smooth $4$ -manifold.

Depends on

Used by

Dependency tree · two levels

46 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

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