Current-compatible signed intersection number

Let $X$ be an oriented smooth $n$-manifold and let $C,D\subseteq X$ be compact oriented embedded smooth submanifolds without boundary, with dimensions $p$ and $q=n-p$. Assume $T_{x}C+T_{x}D=T_{x}X$ at every $x\in C\cap D$. For such $x$, the addition map $A_x:T_{x}C\oplus T_{x}D\to T_{x}X$, $(u,v)\mapsto u+v$, is an isomorphism. Use the determinant-line isomorphism $\det (T_{x}C\oplus T_{x}D)\cong \det (T_{x}C)\otimes \det (T_{x}D)$ in C-first order and let ${\mathrm{\Lambda }}_x:\det (T_{x}C)\otimes \det (T_{x}D)\to \det (T_{x}X)$ be the map induced by $A_x$. Write ${\mathfrak{o}}_{C,x}\subset \det (T_{x}C)$, ${\mathfrak{o}}_{D,x}\subset \det (T_{x}D)$, and ${\mathfrak{o}}_{X,x}\subset \det (T_{x}X)$ for the orientation rays at $x$. For a zero-dimensional factor $M$, identify $\det (T_{x}M)=\mathbb{R}$ using its canonical generator $1$ and take ${\mathfrak{o}}_{M,x}={\mathbb{R}}_{>0}$ when its supplied point sign is $+1$ and ${\mathfrak{o}}_{M,x}={\mathbb{R}}_{<0}$ when the sign is $-1$. Set ${\sigma }_x(C,D)=+1$ if ${\mathrm{\Lambda }}_x({\mathfrak{o}}_{C,x}\otimes {\mathfrak{o}}_{D,x})={\mathfrak{o}}_{X,x}$, and set ${\sigma }_x(C,D)=-1$ if the image is the opposite ray. When $p,q>0$, this agrees with the sign obtained by concatenating positive tangent bases; the zero-dimensional ray uses the supplied point orientation, which is not determined by the empty frame alone. Define the current-compatible intersection number by The sum is finite by the preceding lemma. This convention reverses the usual C-first orientation order when $pq$ is odd; equivalently it is the D-first signed intersection number.

De Rham currents of compact oriented source cycles

Oriented smooth manifolds and oriented charts

Transverse complementary submanifolds meet in finitely many points

The orientation supplier defines each orientation as a smooth choice of a ray in the determinant line of the corresponding tangent space; for positive-dimensional tangent spaces, a basis is positive exactly when its wedge lies in that ray. In the determinant-line definition above, exchanging the $p$- and $q$-dimensional factors changes the induced determinant map by $(-1{)}^{pq}$, so ${\sigma }_x(D,C)=(-1{)}^{pq}{\sigma }_x(C,D)$ in all dimensions, including zero-dimensional factors. Thus the displayed convention is exactly the D-first signed sum. The determinant-line sign is uniquely defined without choosing bases, and the next lemma proves finite-sum well-definedness. For a wrapped source three-cycle and a complementary test three-cycle, $pq=9$, so this current convention is the negative of the convention that orders the source tangent basis before the test tangent basis. The pairing lemma below carries this sign explicitly.