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High normal Pontryagin classes obstruct low-codimension immersions

Statement

Assume AC. Let $M$ be a closed connected smooth $m$ -manifold and let $k \geq 1$ . If ${\overset{ˉ}{p}}_{i} (M) \neq 0$ in $H^{4 i} (M; Q)$ for some $i \geq 1$ with $2 i > k$ (Normal Stiefel-Whitney and Pontryagin classes of a closed manifold), then $M$ does not immerse in $R^{m + k}$ ; equivalently, if ${\overset{ˉ}{p}}_{i} (M) \neq 0$ then every immersion of $M$ has codimension at least $2 i$ , so $M$ does not immerse in $R^{m + 2 i - 1}$ . All assertions are over $Q$ : the integral Whitney product for Pontryagin classes carries a two-torsion correction and the integral form of this test is not asserted.

Facts & Assumptions

Given: A closed connected smooth $m$ -manifold $M$ , an integer $k \geq 1$ , an index $i \geq 1$ with $2 i > k$ and ${\overset{ˉ}{p}}_{i} (M) \neq 0$ in $H^{4 i} (M; Q)$ , and AC (The Axiom of Choice).

[F1]

AC implies the countable choice used by the normal-bundle splitting (AC implies DC implies countable choice).

[F2]

For a smooth immersion $f : M ↬ R^{n}$ of a closed smooth $m$ -manifold with $n > m$ , the normal bundle $ν_{f}$ of rank $n - m$ is a rank- $(n - m)$ stable normal inverse of $M$ , with $T M \oplus ν_{f} ≅ ε^{n}$ (An immersion into R^n gives a rank-(n-m) representative of the stable normal bundle).

[F3]

The normal Pontryagin class is $\overset{ˉ}{p} (M) = p (T M)^{- 1} = p (ν)$ over $Q$ for any stable normal inverse $(ν, φ)$ of $M$ , so $p_{i} (ν) = {\overset{ˉ}{p}}_{i} (M)$ for every $i$ (Normal Stiefel-Whitney and Pontryagin classes of a closed manifold, The normal Pontryagin class is the rational inverse of the tangent Pontryagin class).

[F4]

Pontryagin classes vanish above the rank: if $rank E = r$ then $p_{i} (E) = 0$ for $2 i > r$ (Naturality, stability, and mod-two reduction of Pontryagin classes, Pontryagin classes by complexification).

Proof

1.1F1F2

Suppose for contradiction that $f : M ↬ R^{m + k}$ is a smooth immersion. Since $k \geq 1$ and $M$ is closed, [F2] applies with $n = m + k > m$ under the countable choice granted by [F1]; therefore the normal bundle $ν_{f}$ , of rank $k$ , is a rank- $k$ stable normal inverse of $M$ .

2.1F3F4step 1.1

By [F3] the class $p_{i} (ν_{f}) = {\overset{ˉ}{p}}_{i} (M)$ for the given index $i$ , and $2 i > k = rank ν_{f}$ , so [F4] gives $p_{i} (ν_{f}) = 0$ , contradicting ${\overset{ˉ}{p}}_{i} (M) \neq 0$ . Hence no immersion of $M$ into $R^{m + k}$ exists.

3.1F3step 2.1∎

The final sentence is the case $k = 2 i - 1$ : if ${\overset{ˉ}{p}}_{i} (M) \neq 0$ then no immersion into $R^{m + 2 i - 1}$ exists, because $2 i > 2 i - 1$ . Equivalently every immersion of $M$ has codimension at least $2 i$ under this hypothesis. The argument is stated over $Q$ because the rank vanishing and the inverse identity for $p_{i}$ are used rationally; the integral form is not asserted, since the integral Whitney product for Pontryagin classes only holds modulo two-torsion. No orientation of $M$ is needed beyond the rational normalization.

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Sources

John W. Milnor and James D. Stasheff, Characteristic Classes (Annals of Mathematics Studies 74, Princeton University Press; complete text) (standard reference, not scraped)
Arkadiy Skopenkov, Embedding and Knotting of Manifolds in Euclidean Spaces (arXiv:math/0604045) (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)