A source-cycle current pairs with a transverse dual cycle by intersection number

For $M=C$ and $M=X$, if $M$ is empty then [F12] gives its zero fundamental class and there are no local generators to extend. Otherwise [F9] identifies the integral local generator fixed by every positive smooth chart, or by the supplied point sign when $\dim M=0$. The relative singular chain groups $C_{\ast }(M,M\setminus \{x\};\mathbb{Z})$ are free abelian, with basis the singular simplices not contained in $M\setminus \{x\}$; tensoring these basis groups with $\mathbb{R}$ gives the relative chain complex with real coefficients. Thus [F11] applies with PID $\mathbb{Z}$ and left coefficient module $\mathbb{R}$ under the usual integer action. By [F10], its tensor term is $\mathbb{Z}{\otimes }_{\mathbb{Z}}\mathbb{R}\cong \mathbb{R}$ and its Tor term is ${\mathrm{Tor}}_1^{\mathbb{Z}}(0,\mathbb{R})=0$. Naturality of [F11] under chart restrictions makes the coefficient images of these local generators a continuous real local orientation. By [F12] this gives $[C{]}_{\mathbb{R}}\in H_p(C;\mathbb{R})$ and $[X{]}_{\mathbb{R}}\in H_n(X;\mathbb{R})$; applying [F13] to $(i_C,\mathrm{\varnothing }):(C,\mathrm{\varnothing })\to (X,\mathrm{\varnothing })$ gives $[C{]}_X=(i_C{)}_{\ast }[C{]}_{\mathbb{R}}$. Now [F6], with the stated AC hypothesis and real coefficients, gives the unique $u_C\in H^{n-p}(X;\mathbb{R})$ satisfying $u_C\cap [X{]}_{\mathbb{R}}=[C{]}_X$. This records topological duality only; no identification of current cohomology with singular cohomology is inferred.

By [F5], list the finite intersection as $C\cap D=\{x_1,\dots ,x_m\}$ and choose pairwise disjoint adapted neighborhoods $U_j$ of these points. The set $K=C\setminus {\bigcup }_j{U}_j$ is compact and disjoint from the compact set $D$. By [F7], $K$ and $D$ have disjoint open neighborhoods; together with ${\bigcup }_j{U}_j$ this gives an open neighborhood of all of $C$ whose part away from the $U_j$ misses $D$. Since the supports of ${\alpha }_{ϵ}$ shrink to $C$ by [F4], for sufficiently small $ϵ$ only the neighborhoods $U_j$ contribute, so ${\int }_D{\alpha }_{ϵ}={\sum }_{j=1}^m{\int }_{D\cap U_j}{\alpha }_{ϵ}\mathrm{.}$ The finite chart sum defining ${\alpha }_{ϵ}$ can be evaluated termwise on these neighborhoods.

Fix $x_j$. A chart term from [F4] whose base support omits $x_j$ vanishes near it; for every term whose base support contains $x_j$, [F14] gives ordinary slice coordinates $(u_a,v_a)$ with $C=\{v_a=0\}$ near $x_j$, and choose local coordinates $s_a$ on $D$. Terms with ${\chi }_a(x_j)=0$ are included in the calculation below and have zero local limit. Let ${ϵ}_{C,a},{ϵ}_{D,a},{ϵ}_{X,a}\in \{+1,-1\}$ be the signs of the $u_a$, $s_a$, and $(u_a,v_a)$ coordinate determinant generators relative to the orientations of $C,D,X$, respectively, using point signs for zero-dimensional factors. Put ${\gamma }_a=\mathrm{sgn}\det D_{s_a}(v_a{\mathrm{\mid }}_D)$ and set the determinant of a $0\times 0$ matrix equal to $1$. The differential of $v_a{\mathrm{\mid }}_D$ is injective because its kernel is $T_{x_j}C\cap T_{x_j}D=\{0\}$; source and target both have dimension $q$, so it is an isomorphism by transversality. By [F8], $v_a{\mathrm{\mid }}_D$ is a local diffeomorphism, so it supplies coordinates on $D$ near $x_j$. The block matrix for the addition map $T_{x_j}C\oplus T_{x_j}D\to T_{x_j}X$ is $\left(\begin{array}{cc}I& \ast \\ 0& D_{s_a}(v_a{\mathrm{\mid }}_D)\end{array}\right),$ so the determinant-line definition [F2] gives ${\sigma }_{x_j}(C,D)={ϵ}_{C,a}{ϵ}_{D,a}{ϵ}_{X,a}{\gamma }_a$. The normal factor in [F4] is ${\kappa }_a={ϵ}_{X,a}{ϵ}_{C,a}$, and pulling $d{v}_a^1\wedge \cdots \wedge d{v}_a^q$ back to the oriented $D$ chart contributes ${ϵ}_{D,a}{\gamma }_a$. Changing variables $v_a=ϵz$ cancels the absolute determinant and gives local limit $(-1{)}^{pq}{\kappa }_a{ϵ}_{D,a}{\gamma }_a{\chi }_a(x_j){\int }_{{\mathbb{R}}^q}\rho (z)dz=(-1{)}^{pq}{\sigma }_{x_j}(C,D){\chi }_a(x_j)$. The normalized bump has integral $1$, and ${\sum }_a{\chi }_a(x_j)=1$, so summing all chart terms gives $(-1{)}^{pq}{\sigma }_{x_j}(C,D)$ at this point. The factor $(-1{)}^{pq}$ comes from the current convention $T_{\alpha }(\eta )={\int }_X\alpha \wedge \eta$ in [F1].

Summing the finitely many local limits in step 2.1 gives by [F2]. The normalized partition on $C$ sums to one at every intersection, and every normal bump has integral one, so changing charts, cutoffs, or normalized bumps leaves the same sum. This proves existence and independence of the regularized pairing. If there is exactly one current-positively oriented intersection, the sum is $1$.