Framed regular preimages of a map to a sphere

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix the standard orientation of $S^k$ and identify $S^k=D^k\mathrm{/}{S}^{k-1}$ with the one-point compactification ${\mathbb{R}}^k\cup \{\mathrm{\infty }\}$ of the zero section of the trivial rank-$k$ bundle over a point, so a chosen centre $y_0$ corresponds to $0$ (Trivial Thom spaces as suspension smash products, Disk bundle, sphere bundle, and Thom space: the differential topology interface). Let $X$ be a closed smooth manifold, let $f:X\to S^k$ be smooth and let $y\in S^k$ be a regular value, with $b=(b_1,\dots ,b_k)$ a positively oriented basis of $T_y{S}^k$ (Smooth manifolds and their smooth charts, Transversality to a point is the regular-value condition); read $b$ as the trivialization $\beta :T_y{S}^k\to {\mathbb{R}}^k$ sending $b_i$ to the $i$-th standard basis vector (Orientation of a finite-dimensional real vector space).

Then $N=f^{-1}(y)$ is a closed embedded submanifold of $X$ of codimension $k$, and the differential of $f$ factors through the normal quotient to a smooth bundle isomorphism $\nu (N\subseteq X)⟶(f{\mathrm{\mid }}_N{)}^{\ast }T{S}^k;$ composing it with $\beta$ gives a framing $f_{\ast }b:\nu (N\subseteq X)⟶N\times {\mathbb{R}}^k\mathrm{.}$ The pair $(N,f_{\ast }b)$ is the framed regular preimage of $(f,y,b)$.

Both assertions are the case $E={\mathbb{R}}^k$ of Transverse preimages carry the pulled-back normal structure, applied locally in a target chart centred at $y$, whose differential at $y$ is $\beta$, to the trivial rank-$k$ bundle over the one-point base: its Thom space is $S^k={\mathbb{R}}^k\cup \{\mathrm{\infty }\}$, its zero section is $\{0\}$, transversality of $f$ to $\{0\}$ is exactly regularity of $y$ (Transverse smooth maps, Transversality to a point is the regular-value condition), and clauses (i)-(ii) of that proposition give the closed embedded submanifold and the specified isomorphism of the normal quotient with the pulled-back target fibre; inserting the positive basis $\beta$ turns the target factor into ${\mathbb{R}}^k$ and is precisely the framing. The construction is independent of any auxiliary choice: only the derivative of $f$ along $N$, the value $y$ and the basis $b$ enter, so no chart of $S^k$ at $y$ is chosen and the framing is the composite displayed above. Regular values are not assumed to exist a priori; they are dense by Regular values form a dense $G_{\delta }$ set, and for $k\ge 1$ the framed cobordism class is shown to be independent of the regular value and positive basis in the two following lemmas.

Rank $k=0$ and $N=\mathrm{\varnothing }$ are included: for $k=0$ the sphere is $S^0$, the basis is empty, $\beta$ is the unique map $T_y{S}^0\to {\mathbb{R}}^0$, and $N=f^{-1}(y)$ is a clopen submanifold of $X$ of dimension $\dim X$, carrying the unique rank-zero framing; for $N=\mathrm{\varnothing }$ the normal bundle and the framing are empty. Regular-value independence does not extend to $k=0$: a constant map from a point to $S^0$ has the point and the empty set as its two regular fibres. They are not framed cobordant within the point, since a compact neat codimension-zero submanifold of $I$ is clopen, and one containing $0$ must also contain $1$. The countable-choice hypothesis is inherited exactly from Transverse preimages carry the pulled-back normal structure; the definition itself selects nothing.