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The mod two self-intersection is the top Stiefel-Whitney evaluation

Statement

Assume AC. Let $M$ be a boundaryless smooth $n$ -manifold (not assumed orientable) and let $A^{a} \subseteq M$ be a compact boundaryless embedded submanifold with $2 a = n$ ; write $ν_{A}$ for its normal bundle. Then the mod 2 self-intersection $A \cdot_{2} A \in F_{2}$ of The self-intersection number of a complementary-dimensional oriented submanifold is well defined, depends only on $(A, ν_{A})$ , and satisfies $A \cdot_{2} A = ⟨ w_{a} (ν_{A}), [A] ⟩_{2} \in F_{2},$ where $w_{a} (ν_{A}) \in H^{a} (A; F_{2})$ is the top Stiefel-Whitney class of the rank- $a$ normal bundle and $[A] \in H_{a} (A; F_{2})$ is the mod 2 fundamental class. More generally, for a smooth real rank- $r$ bundle $E \to S$ over a closed smooth $r$ -manifold, and a smooth section $σ$ transverse to the zero section, the zero locus $Z$ is finite and $# Z \equiv ⟨ w_{r} (E), [S] ⟩_{2} (m o d 2)$ and $e_{2} (E) \cap [S] = (i_{Z})_{*} [Z]$ where $e_{2}$ is the canonical mod 2 Euler class. The integral self-intersection number requires an oriented normal bundle and is not asserted here; this proposition is the mod 2 fallback, not an integral substitute.

Facts & Assumptions

Given: The boundaryless $n$ -manifold $M$ (not assumed orientable), the closed embedded $A^{a}$ with $2 a = n$ and its normal bundle $ν_{A}$ , all taken over $F_{2}$ .

[F1]

Over $F_{2}$ every real bundle is canonically oriented, so the mod 2 Euler class $e_{2} (E)$ is defined for every real bundle in the Thom scope, and the mod 2 fundamental class of a closed manifold is the canonical orientation class (Euler class by zero-section pullback of the Thom class).

[F2]

The zero-locus duality admits a mod 2 clause with no orientability hypothesis on the ambient: $e_{2} (E) \cap [M] = (i_{Z})_{*} [Z]$ over $F_{2}$ , where the Koszul sign $(- 1)^{r (n - r)}$ becomes $1$ in characteristic two (The zero locus of a transverse section represents the Euler dual).

[F3]

The mod 2 Euler class equals the top Stiefel-Whitney class: $e_{2} (E) = w_{r} (E)$ for a numerable real rank- $r$ bundle in the Thom scope (The mod-two Euler class is the top Stiefel–Whitney class).

[F4]

The mod 2 intersection number is the parity of the finite transverse count, $I_{2} (g, Z) : = I_{2} (f, Z) = # f^{- 1} (Z) m o d 2$ , and it is homotopy invariant: $I_{2} (F_{0}, Z) = I_{2} (F_{1}, Z)$ (The mod 2 intersection number, The mod 2 intersection number is homotopy invariant).

[F5]

The mod 2 self-intersection is the parity of a small transverse normal push-off count. The push-off lemma's orientation-free clauses give $A \cap A_{s} = φ (Z (s))$ and transversality at these points; hence each zero contributes $1 \in F_{2}$ , including rank zero. No integral determinant sign is needed here (The self-intersection number of a complementary-dimensional oriented submanifold, Normal push-off zeros are the self-intersection points).

[F6]

For every numerable real bundle over an admissible base, $w_{1} (E) = 0$ if and only if $E$ is orientable (The first Stiefel–Whitney class classifies orientability).

Proof

technique · repeat the integral argument in $\mathbb F_2$ coefficients, where every bundle is canonically oriented and every manifold has a mod 2 fundamental class, and identify the mod 2 Euler class with the top Stiefel-Whitney class

1.1F1given

Canonical mod 2 data. By [F1] every real vector bundle carries a canonical $F_{2}$ -orientation, so $ν_{A}$ , $T M$ and the normal Thom class are defined over $F_{2}$ , and the mod 2 fundamental class of the closed manifold $A$ is the canonical orientation class of Fundamental class of a compact oriented manifold applied to that orientation.

2.1F2F4F5step 1.1

The zero-locus duality and the push-off count have mod 2 clauses: the $F_{2}$ case of [F2], together with Normal bundle of the zero locus of a transverse section and [F5], gives for a small transverse section $s$ (whose existence is the finite spanning-section construction in The self-intersection number of a complementary-dimensional oriented submanifold) of $ν_{A} \to A$ that $e_{2} (ν_{A}) \cap [A] = [Z (s)]$ with $Z (s) = A \cap A_{s}$ under the tube; evaluating on the mod 2 fundamental class gives $A \cdot_{2} A = ⟨ e_{2} (ν_{A}), [A] ⟩_{2}$ , and the isotopy argument of [F4] makes the count independent of the push-off.

3.1F2F3F6step 2.1∎

Identify the classes: [F3] states $e_{2} (E) = w_{r} (E)$ for every numerable real rank- $r$ bundle in the Thom scope, and a closed smooth manifold is such a base and its smooth bundles are numerable by Smooth manifolds have CW homotopy type. Substituting into step 2.1 gives the displayed formula $A \cdot_{2} A = ⟨ w_{a} (ν_{A}), [A] ⟩_{2}$ , and the general bundle clause $# Z \equiv ⟨ w_{r} (E), [S] ⟩_{2} (m o d 2)$ with $e_{2} (E) \cap [S] = (i_{Z})_{*} [Z]$ follows from the same proposition. When $ν_{A}$ is nonorientable, [F6] shows that $w_{1} (ν_{A}) \neq 0$ obstructs an integral orientation, and no integral claim is made here. The mod 2 statement is the fallback, not an integral substitute.

Remarks

For oriented $A$ in oriented $M$ , the integral counterpart is The self-intersection number is the Euler number of the normal bundle. The mod two proof above uses the zero-locus supplier directly and does not require that integral counterpart.

Depends on

Used by

Dependency tree · two levels

144 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)
John W. Milnor and James D. Stasheff, Characteristic Classes (Princeton University Press, 1974; complete PDF) (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall, 1974; complete 236-page PDF) (standard reference, not scraped)