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The double point dimension count for surfaces in four- and five-space

Example

Assume ${A C}_{ω}$ . Let $M^{2}$ be a closed connected surface ( $m = 2$ ). Then:

for a self-transverse immersion $f : M^{2} ↬ R^{4}$ the expected dimension is $2 m - n = 0$ : the double point locus is a closed $0$ -dimensional submanifold of $M \times M ∖ Δ_{M}$ , the double point set is finite, and the selected branch pairs are isolated and transverse; several pairs may initially have the same collision image;
for a self-transverse immersion $f : M^{2} ↬ R^{5}$ the expected dimension is $- 1 < 0$ : the double point locus is empty, so $f$ is an injective immersion, and since $M$ is closed $f$ is an embedding.

The example shows why dimension four is the critical case for surfaces: it is exactly there that the double points are isolated rather than absent, so that the algebraic branch-pair count can be nonzero; the $m \geq 3$ disjunction theorem does not cover this surface-in-four-space case. In five-space the configuration space argument already forbids double points.

Facts & Assumptions

Given: Countable choice, a closed connected surface $M$ and a self-transverse immersion $f : M^{2} ↬ R^{n}$ with $n = 4$ or $n = 5$ .

[F1]

For a self-transverse immersion $f : M^{m} \to X^{n}$ , $Δ_{2} (f)$ is a closed embedded submanifold of $M \times M ∖ Δ_{M}$ of pure dimension $2 m - n$ when $2 m - n \geq 0$ , and it is empty when $2 m - n < 0$ ; the double point set is $Σ (f) = f ({pr}_{1} Δ_{2} (f))$ (The double point locus has the expected dimension $2 m - n$ , Self-transverse immersions and the double point locus).

[F2]

A self-transverse immersion $f : M^{m} \to X^{n}$ with $n > 2 m$ is injective (A self-transverse immersion has no double points when $n > 2 m$ ).

[F3]

A proper injective immersion of smooth manifolds is a smooth embedding (A proper injective immersion is a smooth embedding); an immersion is locally an embedding, and a self-transverse immersion is in particular an immersion (Every immersion is locally an embedding, Immersions, submersions, and constant-rank maps).

[F4]

Compact subsets admit finite subcovers from ambient open covers (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it). A closed manifold is compact without boundary, and the diagonal map identifies $Δ_{M}$ homeomorphically with the compact manifold $M$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, The diagonal of a smooth manifold is a closed embedded submanifold); $M \times M$ is compact (A product of finitely many compact spaces is compact in the product topology); a closed subset of a compact space is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

[F5]

An embedded submanifold of dimension $0$ has each point isolated: in a slice chart at the point the submanifold meets the chart in that single point (Embedded submanifolds and slice charts).

[A1]

Countable choice is inherited from the transversality machinery used in [F1]; the finite compactness argument below selects nothing (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

technique · direct

1.1F1F5A1

Clause 1: with $m = 2$ and $n = 4$ the expected dimension is $2 m - n = 0$ , so [F1] makes $Δ_{2} (f)$ a closed embedded $0$ -dimensional submanifold of $M \times M ∖ Δ_{M}$ ; by [F5] each of its points is isolated.

1.2F3F4

A neighbourhood of the diagonal free of double points: by [F3] every $x \in M$ belongs to an open set $U$ on which $f$ is injective. Thus the family of all $U \times U$ with this property is an ambient-open cover of $Δ_{M}$ . By [F4] finitely many $U_{1} \times U_{1}, \dots, U_{k} \times U_{k}$ cover the compact diagonal; their union $N$ is an open neighbourhood of $Δ_{M}$ . If $(y, z) \in N \cap Δ_{2} (f)$ , then $(y, z) \in U_{i} \times U_{i}$ for some $i$ and $f (y) = f (z)$ with $y \neq z$ , contradicting injectivity on $U_{i}$ . Hence $N \cap Δ_{2} (f) = \emptyset$ . No family of neighbourhoods indexed by all source points was selected.

1.3F1F2F3F4

Clause 2: with $m = 2$ and $n = 5$ one has $n > 2 m$ , so [F2] makes $f$ injective; equivalently the expected dimension $2 m - n = - 1$ is negative and [F1] gives $Δ_{2} (f) = \emptyset$ directly. Since $M$ is closed, every compact subset of $R^{5}$ has compact preimage under the continuous map $f$ (the preimage is closed in the compact space $M$ and hence compact by [F4]), so $f$ is proper; being a proper injective immersion, $f$ is a smooth embedding by [F3].

2.1F1step 1.1step 1.2

Finiteness of the double point set: by step 1.2 one has $Δ_{2} (f) \subseteq M \times M ∖ N$ ; the set $M \times M ∖ N$ is closed in $M \times M ∖ Δ_{M}$ because $N$ is open and contains $Δ_{M}$ , and $Δ_{2} (f)$ is closed in $M \times M ∖ Δ_{M}$ by step 1.1, so $Δ_{2} (f)$ is closed in $M \times M ∖ N$ , while $M \times M ∖ N$ is compact by [F4] as a closed subset of the compact space $M \times M$ ; hence $Δ_{2} (f)$ is compact. By step 1.1 it is a discrete subspace, and a compact discrete subspace is finite, so $Σ (f) = f ({pr}_{1} Δ_{2} (f))$ is finite as the image of a finite set. The double points are isolated by step 1.1 and transverse because $f$ is self-transverse by hypothesis and the branch tangents span the target tangent space at each double point.

3.1step 1.1step 1.2step 2.1step 1.3∎

Both clauses hold: clause 1 is steps 1.1, 1.2 and 2.1, and clause 2 is step 1.3. Hence the critical dimension for surfaces is four, where the double point locus is a finite set, while in five-space self-transversality already forces an embedding.

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