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The fundamental class of a boundary pushes forward to zero

Statement

Let $W$ be a compact $R$ -oriented smooth $(n + 1)$ -manifold with boundary $M = \partial W$ , where $R$ is a commutative unital ring (Relative fundamental class and boundary orientation). The manifold $M$ carries the induced boundary orientation, and for $R = F_{2}$ the canonical mod-two orientation may be used, so that the statement applies to every compact smooth manifold (Every manifold is F2-orientable and orientability is componentwise). Let $i : M ↪ W$ be the inclusion, a closed embedding of a smooth $n$ -manifold, and let $[M] \in H_{n} (M; R)$ be the fundamental class of the induced orientation (Fundamental class of a compact oriented manifold).

Then $i_{*} [M] = 0$ in $H_{n} (W; R)$ , and consequently $⟨ i^{*} α, [M] ⟩ = 0 for every α \in H^{n} (W; R) .$ Empty boundary, dimension zero, disconnected manifolds and the empty manifold are included, and no choice principle is used.

Facts & Assumptions

Given: A compact $R$ -oriented smooth $(n + 1)$ -manifold $W$ with boundary $M = \partial W$ , the inclusion $i : M \to W$ , the induced boundary orientation on $M$ , and the fundamental class $[M] \in H_{n} (M; R)$ .

[F1]

The relative fundamental class $[W, M] \in H_{n + 1} (W, M; R)$ is the unique class restricting to the prescribed local generators at interior points, the induced boundary orientation on $M$ is the one whose local generator at $x$ is the restriction of the connector $\partial [W, M]$ , and consequently $\partial [W, M] = [M]$ in $H_{n} (M; R)$ ; the construction handles closed components, dimension zero and the empty case, and uses no AC (Relative fundamental class and boundary orientation, Fundamental class of a compact oriented manifold).

[F2]

The singular homology of the pair $(W, M)$ is naturally long exact: $\dots \to H_{n + 1} (W, M; R) \overset{\partial}{\to} H_{n} (M; R) \overset{i_{*}}{\to} H_{n} (W; R) \overset{j_{*}}{\to} H_{n} (W, M; R) \to \dots$ , so at $H_{n} (M; R)$ the image of $\partial$ equals the kernel of $i_{*}$ (Long exact sequence of a pair).

[F3]

The Kronecker pairing $⟨ \cdot, \cdot ⟩ : H^{n} (W; R) \times H_{n} (W; R) \to R$ descends through cocycle and cycle representatives, is additive in each variable, and satisfies $⟨ f^{*} α, z ⟩ = ⟨ α, f_{*} z ⟩$ for every continuous $f$ (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives).

[F4]

For $R = F_{2}$ every topological manifold carries a canonical $F_{2}$ -orientation, and this construction is choice-free (Every manifold is F2-orientable and orientability is componentwise).

[F5]

If $W$ is compact then its boundary $M$ , being a closed embedded submanifold and hence a closed subset of the compact space $W$ , is compact (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

Proof

1.1F1

( $\partial [W, M] = [M]$ .) By the definition of the relative fundamental class and of the induced boundary orientation, the connector sends the relative fundamental class to the fundamental class of the boundary in that orientation: $\partial [W, M] = [M]$ .

2.1F2step 1.1

( $i_{*} [M] = 0$ .) The long exact homology sequence of the pair $(W, M)$ is exact at $H_{n} (M; R)$ , so the image of $\partial : H_{n + 1} (W, M; R) \to H_{n} (M; R)$ equals the kernel of $i_{*} : H_{n} (M; R) \to H_{n} (W; R)$ . By step 1.1 the class $[M]$ lies in that image, hence $i_{*} [M] = 0$ .

3.1F3step 2.1

(Vanishing of all evaluations.) Let $α \in H^{n} (W; R)$ . Naturality of the Kronecker pairing gives $⟨ i^{*} α, [M] ⟩ = ⟨ α, i_{*} [M] ⟩ = ⟨ α, 0 ⟩ = 0$ by step 2.1 and additivity of the pairing.

4.1F1F4F5step 3.1∎

(Degenerate cases and assembly.) If $\partial W = \emptyset$ then $M = \emptyset$ , the fundamental class $[M]$ is the zero class of $H_{n} (\emptyset; R) = 0$ , and both assertions hold. If $n = 0$ the formula of step 2.1 is the statement that the signed count of the boundary points of a compact oriented one-manifold is zero in $H_{0} (W; R)$ , which is exactly $\partial [W, M] = [M]$ followed by exactness; the earlier steps cover this case without change, as they make no positive-dimensional hypothesis. Disconnected $W$ reduces to the connected case: the finitely many components $W_{λ}$ of $W$ are compact $R$ -oriented manifolds with boundary $\partial W_{λ}$ , the induced boundary orientation of each is the one induced by $W$ , and by the componentwise description of the relative and absolute fundamental classes the boundary class $[M]$ is the finite sum of the images of the classes $[\partial W_{λ}]$ under the inclusions; hence the vanishing proved on each component, together with additivity of $i_{*}$ and of the pairing, gives the assertion for $M$ . For $R = F_{2}$ the induced boundary orientation of the canonical mod-two orientation is the canonical mod-two orientation of $M$ , since over $F_{2}$ each component carries a unique orientation; [F4] and [F5] record the choice-freeness and compactness facts used for this case. No step used a choice principle: the relative fundamental class is unique, exactness and naturality are algebraic, and $M$ is already given as the boundary of $W$ .

Depends on

Used by

Boundaries have zero Stiefel-Whitney numbers Proposition
Oriented boundaries have zero Pontryagin numbers Proposition
Zero-dimensional bordism groups Proposition

Dependency tree · two levels

32 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 (original pagination) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, 2016) (standard reference, not scraped)