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The oriented intersection number reduces to the mod 2 number

Statement

Assume ${A C}_{ω}$ . In the common setting — $X$ compact oriented, $M$ closed oriented, $Z \subseteq M$ closed oriented embedded, $\dim X + \dim Z = \dim M$ — the oriented and mod 2 intersection numbers are related by reduction modulo two: $I (f, Z) \equiv I_{2} (f, Z) (m o d 2)$ for every smooth $f : X \to M$ for which either side is defined. In particular for compact oriented complementary submanifolds $A, B$ , $I (A, B) \equiv I_{2} (A, B) (m o d 2)$ , and $I_{2}$ is defined even where no orientations exist.

Facts & Assumptions

Given: Oriented $X, M, Z$ as in the statement, a smooth map $f : X \to M$ for which either side is defined, and ${A C}_{ω}$ for the transverse-representative selection in 1.1.

[F1]

$I_{2} (f, Z)$ is the parity of the transverse intersection of a transverse representative in the homotopy class of $f$ , and it is independent of that representative (The mod 2 intersection number, The mod 2 intersection number is homotopy invariant).

[F2]

$I (f, Z)$ is the sum of the local signs $ε (p) \in {+ 1, - 1}$ over the finite transverse fibre of a transverse representative, and it is independent of that representative (The oriented intersection number, The oriented intersection number is homotopy invariant).

[F3]

In $Z / 2 Z$ the classes of $+ 1$ and $- 1$ coincide, and reduction of an integer sum is additive (The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[A1]

${A C}_{ω}$ : every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Under it the transversality homotopy theorem supplies, for the given smooth $f$ , a smooth map homotopic to $f$ and transverse to $Z$ (The transversality homotopy theorem).

Proof

technique · direct; reduce both sides to a common transverse representative

1.1A1F1F2given

Choose a smooth map $\tilde{f}$ homotopic to $f$ and transverse to $Z$ ; this is possible by the transversality homotopy theorem under [A1], and by [F1] and [F2] neither $I_{2} (f, Z)$ nor $I (f, Z)$ changes when $f$ is replaced by $\tilde{f}$ . Hence it suffices to prove the congruence for a transverse map, and Countable Choice is used exactly in this selection; the reduction for a transverse map below is choice-free.

2.1F1F2F3step 1.1algebra

For a transverse $\tilde{f}$ the fibre ${\tilde{f}}^{- 1} (Z)$ is finite, $I (\tilde{f}, Z) = \sum_{p} ε (p)$ with $ε (p) = + 1$ or $- 1$ , and $I_{2} (\tilde{f}, Z) = # {\tilde{f}}^{- 1} (Z) m o d 2$ . Each local sign is congruent to $1$ modulo two by [F3], so $I (\tilde{f}, Z) \equiv # {\tilde{f}}^{- 1} (Z) \equiv I_{2} (\tilde{f}, Z) (m o d 2)$ .

3.1F1F2step 2.1∎

For compact oriented complementary submanifolds $A, B$ the inclusion case gives $I (A, B) \equiv I_{2} (A, B)$ ; the mod 2 number of the pair is defined without any orientability hypothesis, so parity remains available without orientations, although comparison with an integer count requires the oriented setting.

Depends on

Used by

A cycle has zero algebraic intersection with a bounding cycle Corollary

Dependency tree · two levels

33 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

Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall, 1974; complete 236-page PDF) (standard reference, not scraped)
John Milnor, Topology from the Differentiable Viewpoint (Princeton University Press; complete 76-page PDF, including the appendix Classifying 1-manifolds) (standard reference, not scraped)