Normalized normal-coordinate forms converge to the source current

If $C=\mathrm{\varnothing }$, take ${\alpha }_{ϵ}=0$; the empty chart sum in [F4] gives ${\int }_C\eta =0$, and the stated limit follows. Assume for the rest of the construction that $C\mathrm{\ne }\mathrm{\varnothing }$. At each point of $C$, choose a nonnegative smooth bump on $C$ supported compactly in an adapted chart's base and positive at that point, using [F3]. Compactness gives finitely many such bumps ${\beta }_j$ whose positive sets cover $C$. Their sum $B={\sum }_j{\beta }_j$ is positive, so ${\chi }_j={\beta }_j\mathrm{/}B$ is a finite smooth partition of unity on $C$ subordinate to the base charts. Each ${\chi }_j$ has compact support in its base chart. Extend it constantly in the normal coordinate near $v=0$.

By [F5], each point of $C$ has an ordinary slice chart because it is an interior point of the boundaryless submanifold; take a finite such cover. For each chart use [F3] in its normal coordinate neighborhood to choose ${\theta }_j\in C_c^{\mathrm{\infty }}({\mathbb{R}}^q)$ supported inside that neighborhood with ${\theta }_j(0)=1$. If $q>0$, put $b(z)=\exp (-1\mathrm{/}(1-\mathrm{\mid }z{\mathrm{\mid }}^2))$ for $\mathrm{\mid }z\mathrm{\mid }<1$ and $b(z)=0$ for $\mathrm{\mid }z\mathrm{\mid }\ge 1$. Every derivative at the unit sphere vanishes because $s^{-N}{e}^{-1\mathrm{/}s}\to 0$ as $s\downarrow 0$ for each $N$, so $b$ is smooth; it is positive on the unit ball and compactly supported. Thus $c_q=\int b(z)dz$ is finite and positive, and $\rho =b\mathrm{/}{c}_q$ is smooth, compactly supported, and has integral one. For $q=0$, take the unit function on ${\mathbb{R}}^0$ with integral one. In the $j$th chart set ${\alpha }_{ϵ,j}=(-1{)}^{pq}{\kappa }_j{\chi }_j(u){\theta }_j(v){ϵ}^{-q}\rho (v\mathrm{/}ϵ)d{v}^1\wedge \cdots \wedge d{v}^q,$ and extend by zero. Compact support of ${\chi }_j$, ${\theta }_j$, and $\rho$ makes this extension smooth for sufficiently small $ϵ$. Set ${\alpha }_{ϵ}={\sum }_j{\alpha }_{ϵ,j}$. Given any open neighborhood $W$ of $C$, each compact set $\mathrm{supp}({\chi }_j)\times \{0\}$ lies inside the chart preimage of $W$. Compactness and openness give ${\delta }_j>0$ such that $\mathrm{supp}({\chi }_j)\times \{\mathrm{\mid }v\mathrm{\mid }\le {\delta }_j\}$ stays inside that preimage. Since $\rho (v\mathrm{/}ϵ)$ is supported where $\mathrm{\mid }v\mathrm{\mid }\le ϵ$, all terms are supported in $W$ for $0<ϵ<{\min }_j{\delta }_j$. Thus this single fixed family has supports shrinking to $C$.

In the $j$th adapted chart, only the tangential coefficient of $\eta$ contributes to ${\alpha }_{ϵ,j}\wedge \eta$. Moving the normal wedge past the $p$ tangential differentials contributes $(-1{)}^{pq}$, which cancels the prefactor. With $v=ϵz$, the signed finite chart integral [F4] is ${\int }_X{\alpha }_{ϵ,j}\wedge \eta ={ϵ}_{X,j}{\kappa }_j\int {\chi }_j(u){\theta }_j(ϵz)\rho (z){\eta }_{\parallel }(u,ϵz)dudz,$ where ${\eta }_{\parallel }$ is the coefficient of $d{u}^1\wedge \cdots \wedge d{u}^p$. On the compact support of ${\chi }_j(u)\rho (z)$, smoothness of $\eta$ and continuity of ${\theta }_j$ give uniform convergence ${\theta }_j(ϵz){\eta }_{\parallel }(u,ϵz)\to {\eta }_{\parallel }(u,0)$. Since ${ϵ}_{X,j}{\kappa }_j={ϵ}_{C,j}$ and $\int \rho =1$, this term converges to ${\int }_C{\chi }_j\eta$, including the orientation sign when a tangential factor has dimension zero. The same formula uses the empty wedge and zero-dimensional integral when $p=0$ or $q=0$.

There are finitely many charts, so summing step 2.1's limits and using ${\sum }_j{\chi }_j=1$ on $C$ gives ${\lim }_{ϵ\downarrow 0}{\int }_X{\alpha }_{ϵ}\wedge \eta ={\int }_C\eta$. By [F1] this is convergence to ${\delta }_C$ on every test form; by [F2] each $T_{{\alpha }_{ϵ}}$ is a current. This proof asserts neither that ${\alpha }_{ϵ}$ is closed nor that it represents a Thom cohomology class.