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Oriented boundaries have zero Pontryagin numbers

Statement

Assume AC (The Axiom of Choice), used only through the Pontryagin class construction, CW-type transport, admissibility input, and the inward field used in the boundary tangent splitting (which requires ${A C}_{ω}$ ). Let $M$ be a closed oriented smooth $4 k$ -manifold that is the boundary of a compact oriented smooth $(4 k + 1)$ -manifold $W$ , in the orientation convention of the null-cobordism definition (Null-cobordant closed manifolds), with fundamental class $[M] \in H_{4 k} (M; Z)$ and inclusion $i : M ↪ W$ . Then $p_{I} [M] = ⟨ p_{i_{1}} \dots p_{i_{r}}, [M] ⟩ = 0$ for every partition $I = (i_{1}, \dots, i_{r})$ of $k$ . Hence a closed oriented $4 k$ -manifold with a nonzero Pontryagin number is not an oriented boundary. The proof uses neither the signature nor the Hirzebruch signature theorem.

Facts & Assumptions

Given: A closed oriented smooth $4 k$ -manifold $M$ occurring as an oriented boundary of a compact oriented $(4 k + 1)$ -manifold $W$ , the inclusion $i : M \to W$ , and the partitions $I$ of $k$ . AC is assumed.

[F1]

$T W ∣_{M} ≅ T M \oplus ε^{1}$ ; the splitting is available under ${A C}_{ω}$ , which AC implies (The boundary stable tangent bundle splits off a trivial line, The Axiom of Choice).

[F2]

On path-connected CW complexes, Pontryagin classes are natural and stable (Naturality, stability, and mod-two reduction of Pontryagin classes). The paragraph “Naturality and stability on CW-type bases” in Pontryagin numbers of a closed oriented manifold derives the same identities for admissible CW-type bases and finite disjoint unions. Both $W$ and $M$ are admissible smooth bases with numerable tangent bundles under AC (Smooth manifolds have CW homotopy type). Thus $p_{i} (i^{*} T W) = i^{*} p_{i} (T W)$ and $p_{i} (T M \oplus ε^{1}) = p_{i} (T M)$ apply here, including disconnected and empty cases.

[F3]

The integral Kronecker pairing is natural and additive: $⟨ i^{*} α, z ⟩ = ⟨ α, i_{*} z ⟩$ (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

For the induced boundary orientation, whose fundamental class is the negative of $[M]$ by the null-cobordism convention, the boundary pushforward vanishes: $i_{*} (- [M]) = 0$ in $H_{4 k} (W; Z)$ (The fundamental class of a boundary pushes forward to zero, Fundamental class of a compact oriented manifold, Null-cobordant closed manifolds).

[F5]

The Pontryagin numbers are defined by $p_{I} [M] = ⟨ p_{i_{1}} \dots p_{i_{r}}, [M] ⟩$ for partitions $I$ of $k$ (Pontryagin numbers of a closed oriented manifold).

Proof

1.1F1F2

(The Pontryagin classes of $M$ are restrictions from $W$ .) By [F1] and stability in [F2], $p (T M) = p (T M \oplus ε^{1}) = p (T W ∣_{M}) = p (i^{*} T W) = i^{*} p (T W),$ so $p_{i_{1}} \dots p_{i_{r}} (T M) = i^{*} (p_{i_{1}} \dots p_{i_{r}} (T W))$ for every partition.

2.1F3F4step 1.1

(The evaluations vanish.) Let $I$ be a partition of $k$ . Naturality of the integral Kronecker pairing [F3], step 1.1, and the vanishing pushforward [F4] give $p_{I} [M] = ⟨ p_{I} (T M), [M] ⟩ = ⟨ i^{*} p_{I} (T W), [M] ⟩ = ⟨ p_{I} (T W), i_{*} [M] ⟩ = 0,$ the last step because $i_{*} [M] = 0$ , which follows from [F4] and linearity of $i_{*}$ .

3.1F5step 2.1∎

(All numbers vanish; the boundary obstruction.) Since the partition $I$ of $k$ was arbitrary, every Pontryagin number of the closed oriented boundary $M$ vanishes. Contrapositively, if a closed oriented $4 k$ -manifold has some nonzero Pontryagin number, it cannot occur as such a boundary. The argument uses only the class naturality and stability, the Kronecker naturality and the boundary pushforward; neither the signature nor the Hirzebruch signature theorem is used.

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Sources

John Milnor and James Stasheff, Characteristic Classes (original pagination) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, 2016) (standard reference, not scraped)