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Embedding obstructions include all immersion normal-class obstructions

Statement

Assume AC. Every smooth embedding is a smooth immersion (Smooth embeddings, Immersions, submersions, and constant-rank maps). Hence, for $k \geq 1$ , the normal-class tests of this page also obstruct embeddings: if $M$ is a closed smooth $m$ -manifold and ${\overset{ˉ}{w}}_{i} (M) \neq 0$ for some $i > k$ , then $M$ admits neither an immersion nor an embedding into $R^{m + k}$ (High normal Stiefel-Whitney classes obstruct low-codimension immersions); if $M$ is connected and ${\overset{ˉ}{p}}_{i} (M) \neq 0$ for some $i$ with $2 i > k$ , then $M$ admits neither an immersion nor an embedding into $R^{m + k}$ (High normal Pontryagin classes obstruct low-codimension immersions). Moreover, for $n \geq m$ , the normal bundle of an embedding into $R^{n}$ is a genuine rank- $(n - m)$ stable normal inverse (for $n > m$ by An embedding into Euclidean space gives a rank-(n-m) stable normal inverse; for $n = m$ , the differential is a fibrewise isomorphism and the normal quotient is the zero bundle), so the rank-vanishing that drives the tests applies to the embedded normal bundle as well. In addition, for $m \geq 1$ and $k \geq 1$ embedding imposes the stronger condition ${\overset{ˉ}{w}}_{k} (T M) = 0$ , and an oriented embedded normal bundle must have $e (ν) = 0$ (Top normal classes vanish for Euclidean embeddings). Thus ${\overset{ˉ}{w}}_{i} (T M) \neq 0$ for any $i \geq k$ obstructs embedding in $R^{m + k}$ ; only $i > k$ is the rank obstruction for immersion. No converse is asserted: these tests are necessary conditions only.

Facts & Assumptions

Given: A closed smooth $m$ -manifold $M$ with $m \geq 1$ , an integer $k \geq 1$ , and AC (The Axiom of Choice).

[F1]

Every smooth embedding is a smooth immersion (Smooth embeddings, Immersions, submersions, and constant-rank maps).

[F2]

For a closed smooth $M$ : if ${\overset{ˉ}{w}}_{i} (M) \neq 0$ for some $i > k$ , then $M$ does not immerse in $R^{m + k}$ (High normal Stiefel-Whitney classes obstruct low-codimension immersions); if $M$ is connected and ${\overset{ˉ}{p}}_{i} (M) \neq 0$ for some $i$ with $2 i > k$ , then $M$ does not immerse in $R^{m + k}$ (High normal Pontryagin classes obstruct low-codimension immersions). These assertions inherit AC from the characteristic-class suppliers; AC also supplies the countable choice required for the normal-bundle splitting (AC implies DC implies countable choice).

[F3]

For $n \geq m$ , the normal bundle of an embedding $M ↪ R^{n}$ is a rank- $(n - m)$ stable normal inverse (by the embedding lemma for $n > m$ , and directly from the fibrewise-isomorphic differential for $n = m$ ) and realizes the normal classes $\overset{ˉ}{w} (M)$ , $\overset{ˉ}{p} (M)$ (An embedding into Euclidean space gives a rank-(n-m) stable normal inverse, Normal Stiefel-Whitney and Pontryagin classes of a closed manifold).

[F4]

If a closed $M^{m}$ with $m \geq 1$ embeds in $R^{m + k}$ with $k \geq 1$ , then ${\overset{ˉ}{w}}_{k} (T M) = 0$ , and if the embedded normal bundle is integrally oriented then its Euler class vanishes (Top normal classes vanish for Euclidean embeddings).

Proof

1.1F1F2

Let $g : M ↪ R^{m + k}$ be a smooth embedding. By [F1] the map $g$ is a smooth immersion, so every non-immersion statement applies to it: in particular, if ${\overset{ˉ}{w}}_{i} (M) \neq 0$ for some $i > k$ , then $M$ has no immersion, hence no embedding, into $R^{m + k}$ ; and if $M$ is connected and ${\overset{ˉ}{p}}_{i} (M) \neq 0$ for some $i$ with $2 i > k$ , then again no immersion and no embedding exists.

1.2F4

The embedding case is in fact stronger in top degree. Let $g : M^{m} ↪ R^{m + k}$ with $m \geq 1$ , $k \geq 1$ . Its normal bundle $ν$ has rank $k$ and, by [F4], satisfies ${\overset{ˉ}{w}}_{k} (T M) = w_{k} (ν) = 0$ , while an integrally oriented embedded normal bundle has $e (ν) = 0$ . Hence if ${\overset{ˉ}{w}}_{i} (T M) \neq 0$ for some $i \geq k$ , then in particular no embedding into $R^{m + k}$ exists, whereas for immersion the rank-vanishing argument only excludes the degrees $i > k$ : the top degree $i = k$ is the embedding-specific condition.

2.1F3step 1.1

The rank-vanishing that drives these tests is realized concretely by the embedding normal bundle: for $n \geq m$ , by [F3] the bundle $ν$ of an embedding $M ↪ R^{n}$ is a genuine rank- $(n - m)$ stable normal inverse, so $w_{i} (ν) = {\overset{ˉ}{w}}_{i} (M)$ and $p_{i} (ν) = {\overset{ˉ}{p}}_{i} (M)$ for every $i$ , and the classes of degree above the rank vanish by the rank convention. Thus the same normal classes carry both the transferred immersion tests and the top-degree embedding test.

3.1F2F3F4step 1.2step 2.1∎

Summarizing, the necessary conditions for an embedding of a closed smooth $M^{m}$ into $R^{m + k}$ with $m \geq 1$ , $k \geq 1$ are: ${\overset{ˉ}{w}}_{i} (M) = 0$ for all $i \geq k$ ; ${\overset{ˉ}{p}}_{i} (M) = 0$ for $2 i > k$ when the normal bundle is rationally considered; and $e (ν) = 0$ for an oriented embedded normal bundle. The first of these contains all the rank obstructions to immersion (degrees $i > k$ ) and adds the embedding-specific top class ${\overset{ˉ}{w}}_{k}$ . No converse is asserted: vanishing of all these classes is far from sufficient for embeddability, as the following boundary remark explains; the companion page separately shows that identical stable normal data do not classify isotopy. AC is inherited from the class suppliers; no additional choice is used.

Depends on

Used by

Characteristic-class vanishing is only necessary for embedding Remark

Dependency tree · two levels

46 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

Arkadiy Skopenkov, Embedding and Knotting of Manifolds in Euclidean Spaces (arXiv:math/0604045) (standard reference, not scraped)
John W. Milnor and James D. Stasheff, Characteristic Classes (Annals of Mathematics Studies 74, Princeton University Press; complete text) (standard reference, not scraped)
Ralph L. Cohen, Immersions of Manifolds and Homotopy Theory (lecture notes, 30 June 2022; complete 46-page text) (standard reference, not scraped)