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The geometric intersection number is the Poincare-dual cup pairing

Statement

Assume AC. Let $M$ be a closed oriented smooth $n$ -manifold and let $A^{a}, B^{b} \subseteq M$ be closed oriented embedded submanifolds with $a + b = n$ . Write $[A] = (i_{A})_{*} [A]_{A}$ , $[B] = (i_{B})_{*} [B]_{B}$ , and $P D [A] = D_{M}^{- 1} ([A])$ , $P D [B] = D_{M}^{- 1} ([B])$ . In the cohomology-first, front-evaluation cap and cup conventions, $I (A, B) = ⟨ P D [A] ⌣ P D [B], [M] ⟩ .$ For nontransverse submanifolds $I (A, B)$ is computed by a transverse map homotopic to $i_{A}$ ; no embedded representative for that map is required. Over $F_{2}$ the same formula holds without orientability. Consequently this geometric number depends only on the represented homology classes, with the factor order of The local oriented intersection sign.

Facts & Assumptions

Given: AC and $M, A, B$ with their orientations and complementary dimensions as in the statement.

[F1]

Under Countable Choice a smooth map is homotopic to a transverse map; the resulting intersection number is well defined and homotopy invariant (The transversality homotopy theorem, The geometric intersection pairing on a closed oriented manifold).

[F2]

Cap is natural and satisfies $(x ⌣ y) \cap c = y \cap (x \cap c)$ . Evaluation of a top-degree cup is therefore evaluation of its second factor on the cap by the first (Cap naturality and projection formula, Kronecker evaluation pairing).

[F3]

For the tangent-first normal Thom extension $α_{B} \in H^{a} (M, M ∖ B; R)$ , its absolute image satisfies $P D [B] = (- 1)^{a b} {\overset{ˉ}{α}}_{B}$ (The normal Thom class realizes the Poincare dual of a closed submanifold).

[F4]

Excision localizes a class supported on finitely many points to disjoint disk pairs; fundamental classes restrict to their prescribed local orientation generators. A normalized Thom class restricts to the normal fibre generator (Excision for singular cohomology, Fundamental class of a compact oriented manifold, Thom class by fiberwise normalization).

[F5]

The local intersection sign compares $d f (T A)$ followed by $T B$ with $T M$ . Complementary transverse preimages are finite for a compact source and closed target (The local oriented intersection sign, Compact transverse complementary intersections are finite).

Proof

technique · use a transverse map on the original source, restrict the normal Thom class of $B$, and compare its local determinant with the ordered intersection sign

1.1F1F2F5givenchoose

By [F1] choose a smooth $f : A \to M$ , homotopic to $i_{A}$ , transverse to $B$ . Then $I (A, B) = I (f, B)$ and $P = f^{- 1} (B)$ is finite by [F5]. Homotopy invariance of homology gives $f_{*} [A]_{A} = [A]$ . By [F2] and $P D [A] \cap [M] = [A]$ , $⟨ P D [A] ⌣ P D [B], [M] ⟩ = ⟨ f^{*} P D [B], [A]_{A} ⟩ .$ This uses the map $f$ on the fixed oriented source $A$ , and does not replace $A$ by its image.

2.1F3F4F5step 1.1algebra

Pull back $α_{B}$ along the map of pairs $(A, A ∖ P) \to (M, M ∖ B)$ to get $β$ . At $p \in P$ , the quotient derivative $q_{p} : T_{p} A \to ν_{B, f (p)}$ is an isomorphism. A normalized tubular chart for $B$ and a local trivialization of its normal bundle give a normal-component map with derivative $q_{p}$ . The inverse function theorem makes it a local diffeomorphism. In the induced normal coordinates the fibre Thom generator pulls back to $σ_{p}$ times the source orientation generator, where $σ_{p}$ is the orientation-ray sign of $q_{p}$ : using the local diffeomorphism as a source chart proves this directly. Excision [F4] and the finite direct sum of the point-supported relative complexes then give $⟨ \overset{ˉ}{β}, [A]_{A} ⟩ = \sum_{p \in P} σ_{p} .$ In dimension zero the same statement is multiplication of the supplied point and normal orientation units, without an inverse-function argument.

3.1F2F3F4F5step 1.1step 2.1algebra∎

Let $u$ be a positive tangent determinant of $B$ and $v$ a positive normal determinant. By tangent-first normal orientation, $(u, v)$ is positive in $M$ . If $d$ is a positive determinant of $T_{p} A$ , then $(u, d f_{p} (d))$ has sign $σ_{p}$ because quotienting its second block gives $q_{p} (d)$ . Swapping the blocks of dimensions $b, a$ shows that $(d f_{p} (d), u)$ has sign $(- 1)^{a b} σ_{p}$ . Thus [F5] gives $ε_{p} = (- 1)^{a b} σ_{p}$ , including the point-ray case. By [F3], $f^{*} P D [B] = (- 1)^{a b} \overset{ˉ}{β}$ . Combining with steps 1.1–2.1 yields the displayed formula. Empty $P$ gives zero on both sides. Over $F_{2}$ the same finite local evaluation applies with every orientation sign equal to one. AC is inherited through transverse representatives, Thom existence and duality.

Depends on

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141 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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy (author draft bookR4) (standard reference, not scraped)
Eleny-Nicoleta Ionel (notes by Andrew Lin), Stanford Math 215B Differential Topology, Winter 2023 (complete 63-page lecture notes) (standard reference, not scraped)
John W. Milnor and James D. Stasheff, Characteristic Classes (Princeton University Press, 1974; complete PDF) (standard reference, not scraped)