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Relative cap and cup evaluation identity

Statement

Let $W$ be a compact oriented $R$ -oriented smooth $n$ -manifold with boundary $M = \partial W$ , let $a \in H^{p} (W, M; R)$ and $x \in H^{n - p} (W; R)$ , and let $[W, M] \in H_{n} (W, M; R)$ be the relative fundamental class of Relative fundamental class and boundary orientation. Then, in the cohomology-first convention of Relative cap products with quotient domains displayed, $⟨ a ⌣ x, [W, M] ⟩ = ⟨ x, a \cap [W, M] ⟩,$ where the left product is the relative/absolute cup product evaluated by relative Kronecker evaluation and the right pairing is absolute Kronecker evaluation.

Facts & Assumptions

Given: A compact oriented $n$ -manifold $W$ with boundary $M = \partial W$ , a relative cohomology class $a \in H^{p} (W, M; R)$ and an absolute class $x \in H^{n - p} (W; R)$ .

[L1]

The cohomology-first cap formula sends a $p$ -cochain $φ$ and an $n$ -simplex $σ$ to $φ \cap σ = φ (σ [0, \dots, p]) σ [p, \dots, n]$ , extended linearly; for $B = \emptyset$ the relative cap product is $H^{p} (X, A; R) \otimes_{R} H_{n} (X, A; R) \to H_{n - p} (X; R)$ , and its descent is proved from the boundary identity and the quotient comparisons (Relative cap products with quotient domains displayed).

[L2]

The singular cup product is $(φ ⌣ ξ) (σ) = φ (σ [0, \dots, p]) ξ (σ [p, \dots, n])$ , $R$ -bilinear on cochains (Singular cup product on cochains).

[L3]

Relative Kronecker evaluation $⟨ -, - ⟩ : H^{k} (X, A; G) \times H_{k} (X, A; Z) \to G$ and the absolute pairing are well defined, biadditive and natural (Relative Kronecker evaluation is well defined, biadditive and natural).

[L4]

The relative fundamental class $[W, M]$ restricts to the given local orientation at every interior point and satisfies $\partial [W, M] = [M]$ (Relative fundamental class and boundary orientation).

[L5]

For a cocycle $φ$ , the cap product satisfies the boundary identity $\partial (φ \cap c) = (- 1)^{p} φ \cap \partial c$ (Cap product boundary identity).

Proof

technique · direct

1.1L1L2given

Choose a relative $p$ -cocycle $α$ representing $a$ (vanishing on $C_{*} (M)$ ), an absolute $(n - p)$ -cocycle $ξ$ representing $x$ , and a relative $n$ -cycle represented by $c$ for $[W, M]$ . For these representatives, the front/back formulas of [L1] and [L2] give $(α ⌣ ξ) (c) = \sum_{σ} a_{σ} α (σ [0, \dots, p]) ξ (σ [p, \dots, n]) = ξ (α \cap c)$ , an identity of cochains on $W$ .

2.1step 1.1L1L5

If $α$ is a relative cocycle and $c$ a relative $n$ -cycle with $\partial c \in C_{n - 1} (M)$ , then $\partial (α \cap c) = (- 1)^{p} α \cap \partial c$ by [L5], and this vanishes because $α$ vanishes on chains in $M$ ; moreover replacing $c$ by $c + \partial b + c_{M}$ or $α$ by $α + δ u$ changes $α \cap c$ by an absolute boundary, so $α \cap c$ determines a well-defined class $a \cap [W, M] \in H_{n - p} (W; R)$ .

3.1step 2.1L2L3

The cochain $α ⌣ ξ$ is a relative cocycle and its class is the relative/absolute cup product $a ⌣ x$ , so by the well-definedness of relative evaluation [L3] the left side of the statement is $ξ (α \cap c)$ for any representatives; by step 1.1 this equals the absolute evaluation on the right, and step 2.1 shows the right side is the class of $α \cap c$ .

4.1step 3.1L1L3L4∎

Therefore $⟨ a ⌣ x, [W, M] ⟩ = ⟨ x, a \cap [W, M] ⟩$ ; the conventions are exactly the cohomology-first ones fixed in [L1] and [L3], with no extra sign, and the degenerate cases $p = 0$ , $p = n$ , $M = \emptyset$ or $a = 0$ follow from the same computation.

Depends on

Used by

Dependency tree · two levels

20 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

Allen Hatcher, Algebraic Topology, Cambridge University Press 2002 (complete book) (standard reference, not scraped)
Glen E. Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Chapter VI (relative cap and cup products) (standard reference, not scraped)