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The signature theorem imposes divisibility constraints on Pontryagin numbers

Statement

Assume AC, inherited from the Hirzebruch signature theorem. For every closed oriented smooth manifold $M$ of dimension $4 k$ the L-genus value $σ (M) = L [M] = ⟨ L_{k} (T M), [M] ⟩$ is an integer and is the rational polynomial in the Pontryagin numbers of $M$ determined by $L_{k}$ . In low dimensions: (1) every closed oriented $4$ -manifold satisfies $3 ∣ p_{1} [M]$ ; (2) every closed oriented $8$ -manifold satisfies $45 ∣ (7 p_{2} [M] - p_{(1, 1)} [M])$ , where $p_{(1, 1)} [M] = ⟨ p_{1} (T M)^{2}, [M] ⟩$ ; (3) more generally $L_{k} (T M)$ is a rational polynomial in the Pontryagin classes and the integrality of its evaluation on $[M]$ is the arithmetic constraint recorded here, for every $k \geq 0$ . No integrality or divisibility is asserted for the analogous expressions for arbitrary bundles.

Facts & Assumptions

Given: AC; a closed oriented smooth manifold $M$ of dimension $4 k$ ; the tangent Pontryagin classes $p_{i} = p_{i} (T M)$ and the L-polynomial $L_{k}$ .

[F1]

The signature theorem gives $σ (M) = L [M] = ⟨ L_{k} (T M), [M] ⟩$ for every closed oriented smooth $4 k$ -manifold (The Hirzebruch signature theorem).

[F2]

The total L-class is $L (M) = \sum_{j \geq 0} L_{j} (T M)$ , a polynomial in the Pontryagin classes with rational coefficients; explicitly $L_{1} = p_{1} / 3$ and $L_{2} = (7 p_{2} - p_{1}^{2}) / 45$ (The total L-class and the L-genus of a smooth manifold, The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

[F3]

The Pontryagin numbers $p_{I} [M] = ⟨ \prod p_{i_{j}}, [M] ⟩$ are integers and the Kronecker pairing is linear over $Q$ in its cohomology variable (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

[F4]

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

[F5]

The low-dimensional cases are already established: $3 ∣ p_{1} [M]$ for closed oriented $4$ -manifolds and $45 ∣ (7 p_{2} [M] - p_{(1, 1)} [M])$ for closed oriented $8$ -manifolds (The four-dimensional signature formula, The eight-dimensional signature formula).

Proof

technique · direct; specialise the L-polynomial to each degree and read off the integrality constraint

1.1givenF1F2F3F4

By [F1] and [F2], $σ (M) = ⟨ L_{k} (T M), [M] ⟩$ is the evaluation of the rational polynomial $L_{k}$ in the Pontryagin classes on the fundamental class; by [F3] this evaluation is the corresponding rational linear combination of the Pontryagin numbers $p_{I} [M]$ , and by [F4] its value is an integer.

1.2F2F3F4F5

In dimension four, [F2] gives $L_{1} = p_{1} / 3$ , so $3 σ (M) = p_{1} [M]$ ; both sides are integers by [F4] and [F3], hence $3 ∣ p_{1} [M]$ , in agreement with the first clause of [F5].

1.3F2F3F4F5

In dimension eight, [F2] gives $L_{2} = (7 p_{2} - p_{1}^{2}) / 45$ , so $45 σ (M) = 7 p_{2} [M] - p_{(1, 1)} [M]$ ; both sides are integers by [F4] and [F3] applied to the products $p_{1} ⌣ p_{1}$ and $p_{2}$ , hence $45 ∣ (7 p_{2} [M] - p_{(1, 1)} [M])$ , in agreement with the second clause of [F5].

2.1step 1.1F1F2F3F4

In general degree $4 k$ , [F2] makes $L_{k} (T M)$ a rational polynomial in the Pontryagin classes, [F3] turns its evaluation on $[M]$ into the corresponding rational combination of Pontryagin numbers, and [F1] and [F4] identify that combination with the integer $σ (M)$ ; thus for every $k \geq 0$ the integrality of $⟨ L_{k} (T M), [M] ⟩$ is exactly the arithmetic constraint stated in clause (3), and no further arithmetic conclusion is drawn.

3.1givenstep 2.1∎

Scope of the assertion: the identification of $L [M]$ with an integer uses the tangent bundle and the fundamental class of a closed oriented manifold, so nothing here asserts integrality or divisibility for $L$ of an arbitrary real vector bundle; the final sentence of the statement is this scope boundary, not a vanishing claim.

Depends on

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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)