The framing sign of a zero-dimensional regular preimage

Assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$, inherited from the framed preimage of Framed regular preimages of a map to a sphere. Let $M$ be a closed oriented smooth $m$-manifold with $m\ge 1$, and let $(N,\phi )$ be a closed framed $0$-dimensional submanifold of $M$ in the sense of Framings of a normal bundle. Since $\dim N=0$, the normal bundle of $N$ in $M$ is $\nu (N\subseteq M)={\coprod }_{x\in N}{T}_{x}M\mathrm{/}{T}_{x}N={\coprod }_{x\in N}{T}_{x}M$ over the finite set $N$ (Normal and conormal bundles of an embedded submanifold), and the framing is a family of linear isomorphisms ${\phi }_x:T_{x}M\to {\mathbb{R}}^m$; the pair $(x,{\phi }_x)$ is a framing of the point $x$ in the frame-bundle dictionary of The frame bundle of a smooth manifold, namely the inverse of the element $(x,{\phi }_x^{-1})\in B(M{)}_x$.

The framing sign of $x\in N$ is $\epsilon (x):=\{\begin{array}{ll}+1,& {\phi }_x\text{ is orientation-preserving for the given orientation of }{T}_{x}M\text{ and the standard orientation of }{\mathbb{R}}^m,\\ -1,& \text{otherwise},\end{array}$ and the signed count of $(N,\phi )$ is $\mathrm{\Phi }(N,\phi ):={\sum }_{x\in N}\epsilon (x)\in \mathbb{Z}$. Replacing ${\phi }_x$ by $A\circ {\phi }_x$ for $A\in {\mathrm{G}\mathrm{L}}_m(\mathbb{R})$ multiplies $\epsilon (x)$ by the sign of $\det A$, so the sign records exactly the orientation class of the framing and is constant on the two components of the fibre $B(M{)}_x$; when $M$ is oriented and a positive chart is used, $\epsilon (x)=+1$ precisely for the positively oriented framings of Oriented smooth manifolds and oriented charts and Orientation of a finite-dimensional real vector space. The empty $0$-manifold has signed count $0$, and the definition uses no choice beyond the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ and no orientation when only the unframed parity of $N$ is considered.

For the framed regular preimage of a smooth map $f:M\to S^m$ at a regular value $y$ with a positive basis $b$ of $T_y{S}^m$, write $\beta :T_y{S}^m\to {\mathbb{R}}^m$ for the coordinate isomorphism sending the positive basis $b$ to the standard basis. The induced framing is $f_{\ast }b=\beta \circ d{f}_x$ on $\nu (x)=T_{x}M$ (Framed regular preimages of a map to a sphere), and the framing sign of $x\in f^{-1}(y)$ is exactly the local orientation sign $\mathrm{sgn}(d{f}_x)$ of Local orientation sign of a regular preimage: both compare the isomorphism $d{f}_x:T_{x}M\to T_y{S}^m$ of oriented vector spaces, and the positive coordinate isomorphism $\beta$ carries the given orientation of $T_y{S}^m$ to the standard orientation of ${\mathbb{R}}^m$. In particular the signed count of the framed preimage is the sum of the local orientation signs of $f$ over the regular fibre.