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Parallelizable manifolds have no stable characteristic-class obstruction to Euclidean immersion

Statement

Assume AC. Let $M$ be a closed smooth $m$ -manifold whose tangent bundle is trivial, $T M ≅ ε^{m}$ . Then for every $k \geq 1$ the trivial bundle $ε^{k}$ , together with the composite isomorphism $T M \oplus ε^{k} ≅ ε^{m} \oplus ε^{k} ≅ ε^{m + k}$ , is a rank- $k$ stable normal inverse of $M$ ; consequently $M$ admits an immersion into $R^{m + k}$ for every $k \geq 1$ , in particular into $R^{m + 1}$ (Smale-Hirsch makes rank reduction sufficient for Euclidean immersion in positive codimension). Moreover $\overset{ˉ}{w} (M) = 1$ and $\overset{ˉ}{p} (M) = 1$ (Normal Stiefel-Whitney and Pontryagin classes of a closed manifold): the inverse total classes of the trivial tangent class are trivial, so every immersion and embedding test on this page based on ${\overset{ˉ}{w}}_{i}$ or ${\overset{ˉ}{p}}_{i}$ returns zero for $M$ . The proposition asserts nothing about embeddability, about the minimal immersion dimension below $m + 1$ , or about other obstructions; dimension, rank and embedding-theoretic issues are unaffected.

Facts & Assumptions

Given: A closed smooth $m$ -manifold $M$ with a trivialization $T M ≅ ε^{m}$ , and AC (The Axiom of Choice).

[F1]

A rank- $k$ stable normal inverse of $M$ is a smooth real bundle $ν$ of rank $k$ together with a smooth bundle isomorphism $φ : T M \oplus ν \to ε^{m + k}$ (Stable normal inverse of the tangent bundle, Smooth vector bundles, rank, fibres, and trivial bundles, Whitney sums of vector bundles).

[F2]

Under the countable choice ${A C}_{ω}$ (implied by AC by AC implies DC implies countable choice), a rank- $k$ stable normal inverse of a closed $M$ with $k \geq 1$ produces an immersion into $R^{m + k}$ (Smale-Hirsch makes rank reduction sufficient for Euclidean immersion in positive codimension, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

The normal classes of a closed $M$ are $\overset{ˉ}{w} (M) = w (T M)^{- 1}$ in $H^{*} (M; F_{2})$ and $\overset{ˉ}{p} (M) = p (T M)^{- 1}$ in $H^{*} (M; Q)$ for connected $M$ , realized as $w (ν)$ , $p (ν)$ for any stable normal inverse $(ν, φ)$ (Normal Stiefel-Whitney and Pontryagin classes of a closed manifold, The normal Stiefel-Whitney class is the multiplicative inverse of the tangent class, The normal Pontryagin class is the rational inverse of the tangent Pontryagin class).

[F4]

The trivial bundle has trivial characteristic classes: $w (ε^{r}) = 1$ and $p (ε^{r}) = 1$ (The normal Stiefel-Whitney class is the multiplicative inverse of the tangent class, The normal Pontryagin class is the rational inverse of the tangent Pontryagin class, Smooth vector bundles, rank, fibres, and trivial bundles).

Proof

1.1F1

Fix a trivialization $τ : T M \to ε^{m}$ and let $k \geq 1$ . Let $φ : T M \oplus ε^{k} \to ε^{m + k}$ be the composite of $τ \oplus {id}_{ε^{k}} : T M \oplus ε^{k} \to ε^{m} \oplus ε^{k}$ with the canonical associativity identification $ε^{m} \oplus ε^{k} = ε^{m + k}$ . This is a smooth bundle isomorphism, so by [F1] the pair $(ε^{k}, φ)$ is a rank- $k$ stable normal inverse of $M$ .

2.1F2step 1.1

By [F2] and the triviality of AC implies ${A C}_{ω}$ , the rank- $k$ stable normal inverse of step 1.1 produces an immersion $M ↬ R^{m + k}$ for every $k \geq 1$ ; taking $k = 1$ gives an immersion into $R^{m + 1}$ , and any larger codimension is obtained by stabilizing or by the same construction.

2.2F3F4step 1.1

The characteristic classes vanish in the normal direction. Since $T M ≅ ε^{m}$ , naturality of the characteristic classes gives $w (T M) = w (ε^{m}) = 1$ and $p (T M) = p (ε^{m}) = 1$ by [F4]. By [F3] the normal classes are the inverses of these units, so $\overset{ˉ}{w} (M) = 1^{- 1} = 1$ and $\overset{ˉ}{p} (M) = 1^{- 1} = 1$ ; equivalently, the inverse bundle of step 1.1 is trivial, $w (ε^{k}) = 1$ , $p (ε^{k}) = 1$ , and the inverse lemmas identify these with the normal classes. Hence every normal Stiefel-Whitney class ${\overset{ˉ}{w}}_{i} (M)$ with $i \geq 1$ and every normal Pontryagin class ${\overset{ˉ}{p}}_{i} (M)$ with $i \geq 1$ vanishes, so no immersion or embedding test of this page based on $\overset{ˉ}{w}$ or $\overset{ˉ}{p}$ obstructs anything for $M$ .

3.1F1F2F3step 1.1step 2.2∎

Parallelizability supplies a rank- $k$ inverse with trivial normal bundle for every $k \geq 1$ , hence the stated Euclidean immersions by [F2], while the positive normal characteristic classes vanish by step 2.2. AC is inherited from the characteristic-class suppliers and implies the countable choice used by the immersion-existence supplier. No embedding or minimal-dimension conclusion is asserted.

Depends on

Used by

Dependency tree · two levels

48 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

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)
Arkadiy Skopenkov, Embedding and Knotting of Manifolds in Euclidean Spaces (arXiv:math/0604045) (standard reference, not scraped)