Framings of a normal bundle

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $S\subseteq X$ be a closed embedded smooth submanifold of a smooth manifold $X$ without boundary, with normal bundle $\nu (S)=TX{\mathrm{\mid }}_S\mathrm{/}TS$ of rank $k$, a smooth vector bundle over $S$ by Assuming countable choice, normal and conormal bundles are smooth vector bundles (Normal and conormal bundles of an embedded submanifold, Smooth embeddings).

A framing of $S$ in $X$ is a smooth bundle isomorphism $\phi :\nu (S)⟶S\times {\mathbb{R}}^k$ over ${\mathrm{i}\mathrm{d}}_S$ (Smooth vector bundles, rank, fibres, and trivial bundles); equivalently a trivialization of $\nu (S)$, equivalently a global frame of its sections, and in Milnor's metric language the same thing as a framing of the orthogonal complement $T{S}^{\perp }$. A framed submanifold is a pair $(S,\phi )$. When a Riemannian metric on $X$ is supplied, the orthogonal-complement model of the normal bundle is canonically identified with $\nu (S)$ by Assuming countable choice, an ambient metric identifies the two normal bundles, which is how Milnor's framings are read in the metric-free quotient convention used here; a change of ambient metric changes that comparison but not the framing data itself.

Rank $k=0$ and $S=\mathrm{\varnothing }$ are included (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right): a framing of a rank-zero bundle is the unique bundle isomorphism onto $S\times {\mathbb{R}}^0=S$, and the empty framing is unique. A framing is an actual trivialization of $\nu (S)$, never merely a stable isomorphism of it: adding trivial summands to a normal bundle is a different construction, and the stable and unstable notions are kept apart throughout (Smooth vector bundles, rank, fibres, and trivial bundles).

The countable-choice hypothesis is inherited solely from the smooth normal-bundle structure of Assuming countable choice, normal and conormal bundles are smooth vector bundles; this definition itself selects nothing and proves nothing. The symbol $\nu (S)$ always denotes the quotient normal bundle $TX{\mathrm{\mid }}_S\mathrm{/}TS$, and the rank-$k$ trivialization that fixes the identification of the normal fibres with ${\mathbb{R}}^k$ is part of the data, not a choice made afterwards.