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The signature of an oriented boundary vanishes, so the signature is an oriented cobordism invariant

Statement

Assume AC, inherited from Poincare-Lefschetz duality. Let $W$ be a compact oriented smooth $(4 k + 1)$ -manifold with boundary $M = \partial W$ carrying the induced orientation and inclusion $i : M ↪ W$ . Then $σ (M) = 0$ . Consequently, if $M_{0}, M_{1}$ are closed oriented $4 k$ -manifolds that are oriented cobordant (Oriented smooth cobordism), then $σ (M_{0}) = σ (M_{1})$ : by Oriented smooth cobordism, $\partial V = (- M_{0}) ⊔ M_{1}$ ; reversing the orientation of $V$ gives $\partial (- V) = M_{0} ⊔ (- M_{1})$ as an oriented boundary, so $0 = σ (M_{0} ⊔ (- M_{1})) = σ (M_{0}) - σ (M_{1})$ by The signature is additive under disjoint union and negates under orientation reversal. Hence $σ$ descends to a well-defined additive map on oriented bordism classes $Ω_{4 k}^{S O} \to Z$ and vanishes on null-cobordant classes (Null-cobordant closed manifolds).

Facts & Assumptions

Given: AC; a compact oriented smooth $(4 k + 1)$ -manifold $W$ with boundary $M = \partial W$ and inclusion $i$ , with the induced boundary orientation.

[F1]

For a closed oriented $4 k$ -manifold the signature is the inertia difference $σ (M) = p - q$ of the nondegenerate middle form $Q_{M}$ (The signature of a closed oriented manifold of dimension divisible by four, The middle-dimensional intersection form of a closed oriented 4k-manifold).

[F2]

With $K = im (i^{*} : H^{2 k} (W; R) \to H^{2 k} (M; R))$ , one has $K = K^{⊥_{Q_{M}}}$ , and $K$ is a totally isotropic subspace of half the dimension of $H^{2 k} (M; R)$ (The restriction image on a cobordism boundary is Lagrangian).

[F3]

A nondegenerate symmetric form with a totally isotropic subspace of exactly half the dimension has zero signature (A nondegenerate symmetric form with a half-dimensional isotropic subspace has zero signature).

[F4]

The signature is additive over disjoint unions and negates under orientation reversal: $σ (X ⊔ Y) = σ (X) + σ (Y)$ and $σ (- X) = - σ (X)$ (The signature is additive under disjoint union and negates under orientation reversal).

[F5]

Oriented cobordism is the equivalence relation generated by oriented bordisms: for a bordism $V$ from $M_{0}$ to $M_{1}$ , $\partial V = (- M_{0}) ⊔ M_{1}$ , and a null-cobordant closed manifold bounds a compact oriented manifold (Oriented smooth cobordism, Null-cobordant closed manifolds, Unoriented and oriented bordism groups).

Proof

technique · direct; the boundary restriction image is a Lagrangian subspace, so the isotropic lemma kills the signature

1.1givenF1F2F3

Boundary vanishing: by [F2] the subspace $K$ of $H^{2 k} (M; R)$ is totally isotropic for the nondegenerate form $Q_{M}$ and has dimension one half of $\dim H^{2 k} (M; R)$ ; hence $σ (M) = 0$ by [F1] and [F3].

2.1step 1.1F4F5

Cobordism invariance: let $V$ be an oriented cobordism from $M_{0}$ to $M_{1}$ , so that $\partial V = (- M_{0}) ⊔ M_{1}$ by [F5]. Reversing the orientation of $V$ makes its boundary $M_{0} ⊔ (- M_{1})$ , so step 1.1 gives $0 = σ (M_{0} ⊔ (- M_{1})) = σ (M_{0}) + σ (- M_{1}) = σ (M_{0}) - σ (M_{1})$ by [F4]; hence $σ (M_{0}) = σ (M_{1})$ .

3.1step 1.1step 2.1F4F5∎

Descent: steps 1.1 and 2.1 show that $σ$ is constant on oriented cobordism classes and vanishes on null-cobordant manifolds (which bound by [F5]); with additivity [F4] it descends to a well-defined additive map $Ω_{4 k}^{S O} \to Z$ . The empty manifold has $σ (\emptyset) = 0$ and for $k = 0$ the boundary of an oriented 1-manifold has signed count zero, consistent with step 1.1.

Depends on

Used by

The Hirzebruch signature theorem Theorem

Dependency tree · two levels

56 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)