Stable normal inverse of the tangent bundle

Let $M$ be a smooth $m$-manifold (Smooth manifolds and their smooth charts). A stable normal inverse of the tangent bundle is a pair $(\nu ,\phi )$ consisting of a smooth real vector bundle $\nu \to M$ of finite rank $k$ (Smooth vector bundles, rank, fibres, and trivial bundles) and a smooth bundle isomorphism $\phi :TM\oplus \nu ⟶{\epsilon }^{m+k}$ onto the trivial real bundle ${\epsilon }^{m+k}=M\times {\mathbb{R}}^{m+k}$ (The tangent bundle as a disjoint union, Whitney sums of vector bundles, Bundle maps, sections, subbundles, and isomorphisms). A rank-$k$ stable normal inverse is one whose bundle has rank $k$; when the rank is not named, $k=\mathrm{rank}\nu$. Two stable normal inverses $({\nu }_0,{\phi }_0)$, $({\nu }_1,{\phi }_1)$ are stably equivalent when ${\nu }_0\oplus {\epsilon }^a\cong {\nu }_1\oplus {\epsilon }^b$ for some $a,b\ge 0$. Adding a trivial summand, $(\nu ,\phi )\mapsto (\nu \oplus {\epsilon }^1,\phi \oplus {\mathrm{id}}_{{\epsilon }^1})$, carries rank-$k$ inverses to rank-$(k+1)$ inverses and preserves stable equivalence.

This is the inverse-bundle form of the published stable normal bundle Stable normal bundle of a compact smooth manifold: under ${\mathrm{A}\mathrm{C}}_{\omega }$, for compact $M$ and a smooth embedding $i:M\hookrightarrow {\mathbb{R}}^N$ with $N\ge m$, the normal bundle gives a rank-$(N-m)$ example with $TM\oplus {\nu }_i\cong {\epsilon }^N$ (the $N>m$ case is An embedding into Euclidean space gives a rank-(n-m) stable normal inverse ↗; for $N=m$, $di$ is a fibrewise isomorphism and the normal quotient is the zero bundle). The dimension qualification is needed for an empty source, whose fibrewise immersion condition alone imposes no dimension inequality. Under ${\mathrm{A}\mathrm{C}}_{\omega }$ every closed smooth $M$ has such an inverse by An embedding into Euclidean space gives a rank-(n-m) stable normal inverse ↗. The converse statement that every stable normal inverse is stably isomorphic to the normal bundle of an embedding is the stable classification statement; it is neither asserted nor used on this page. No choice principle is part of the definition; ${\mathrm{A}\mathrm{C}}_{\omega }$ enters only through the metric and tubular identifications of the cited embedding lemma.