LemmaStatement: Literature-sourcedProof: AI-adapted

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A self-transverse immersion has no double points when $n > 2 m$

Statement

Assume ${A C}_{ω}$ . Let $f : M^{m} \to X^{n}$ be a self-transverse immersion with $n > 2 m$ . Then $Δ_{2} (f) = \emptyset$ , so $f$ is injective. No properness or compactness of $M$ is used; in particular self-transversality, not genericity, is the hypothesis.

Facts & Assumptions

Given: Countable choice and a self-transverse immersion $f : M^{m} \to X^{n}$ with $n > 2 m$ .

[F1]

$Δ_{2} (f) = {(x, y) \in M \times M ∖ Δ_{M} : f (x) = f (y)}$ and $Σ (f) = f ({pr}_{1} Δ_{2} (f))$ (Self-transverse immersions and the double point locus).

[L1]

For a self-transverse immersion $f : M^{m} \to X^{n}$ , if $2 m - n < 0$ then $Δ_{2} (f) = \emptyset$ and $Σ (f) = \emptyset$ (The double point locus has the expected dimension $2 m - n$ ).

[L2]

Transverse maps $F : U^{x} \to W^{w}$ , $G : Z^{z} \to W^{w}$ with $x + z < w$ have empty fibre product; in particular transverse embedded submanifolds whose dimensions sum to less than the ambient dimension do not meet (Negative expected dimension forces empty generic intersections).

[A1]

Countable choice is inherited from the transversality machinery used in [L1] and [L2]; this proof selects nothing (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1L1L2A1

The hypothesis $n > 2 m$ is $2 m - n < 0$ , so clause 2 of [L1] applies to the self-transverse immersion $f$ and gives $Δ_{2} (f) = \emptyset$ and $Σ (f) = \emptyset$ . The negative expected dimension is the instance of [L2] for the transverse pair $(f \times f, Δ_{X} ↪ X \times X)$ , whose source dimensions $2 m$ and $n$ sum to less than the target dimension $2 n$ exactly when $2 m < n$ .

2.1F1step 1.1

By [F1] an element of $Δ_{2} (f)$ is a pair of distinct points with equal image; since $Δ_{2} (f) = \emptyset$ there are no such pairs, hence $f (x) = f (y)$ implies $x = y$ , that is, $f$ is injective.

3.1step 1.1step 2.1∎

Therefore every self-transverse immersion from an $m$ -manifold into an $n$ -manifold with $n > 2 m$ is injective, without any compactness or properness hypothesis and with self-transversality in place of genericity.

Depends on

Used by

Dependency tree · two levels

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A self-transverse immersion has no double points when n>2mn>2m