Finite chart localization of compactly supported top forms, including boundary

For each $p\in K$, choose a connected chart neighborhood $U_p$ whose coordinate image is a Euclidean ball or a ball intersected with $H^n$, and whose closure lies in a slightly larger chart domain. In the boundary case center the ball at the face point; a small ball intersection with $H^n$ is connected. Apply [F4] to the singleton coordinate point inside a still smaller Euclidean ball and restrict the resulting Euclidean bump to the chart image. Its support there is a closed subset of a compact closed ball or half-ball by [F5]. Pull it back and extend it by zero outside $U_p$. Since its compact support lies in $U_p$, the extension is smooth across every artificial chart edge; at a genuine boundary point it is smooth by restriction of the Euclidean bump. Thus there is a smooth $b:M\to [0,1]$ with $b(p)=1$ and compact support in $U_p$. For $n=0$, each chart is a singleton and its indicator is the required smooth compactly supported bump. This construction asserts existence for each fixed point, without selecting a family indexed by all points.

Let $\mathcal{B}$ be the set of all such chart-and-bump tuples $(U,\varphi ,b)$, and put $V_b=\{b>1\mathrm{/}2\}$. These open sets cover $K$. Compactness in [F5] supplies finitely many tuples with $K\subseteq {\bigcup }_{i=1}^m{V}_{b_i}$. Put $B={\sum }_i{b}_i$, $\theta =s(4B-1)$, and define ${\chi }_i=\theta b_i\mathrm{/}B$ where $B>0$, and $0$ where $B=0$. The quotient is smooth because $\theta =0$ on the open set $B<1\mathrm{/}4$, including a neighbourhood of every zero of $B$. Its support lies in the compact support of $b_i$, and ${\sum }_i{\chi }_i=\theta =1$ on the open set $B>1\mathrm{/}2$ containing $K$. All ${\chi }_i$ are nonnegative. For $K=\mathrm{\varnothing }$ use the empty family.

We first compare two charts for a top form $\zeta$ whose compact support $L$ lies in their overlap. At each point of $L$, choose a relatively open overlap neighbourhood on which the transition is a Euclidean diffeomorphism if the point is interior, or has a side-preserving Euclidean extension if it lies on the genuine face. Shrink the neighbourhood so that its closure is still inside the extension domain and both original charts. Apply the finite construction of steps 1.1 and 2.1 to this set of eligible neighbourhoods, obtaining finitely many nonnegative ${\lambda }_j$ with compact support in individual transition neighbourhoods and ${\sum }_j{\lambda }_j=1$ near $L$. Thus $\zeta ={\sum }_j{\lambda }_j\zeta$ is a finite identity. This uses all eligible local tuples followed by a finite subcover, not a global partition theorem.

For one summand in step 3.1 write $G=\psi {\varphi }^{-1}$ and its two coefficients as $f_x=(f_y\circ G)\det DG$ on the half-space overlap. The chart signs satisfy ${\sigma }_{\varphi }\det DG={\sigma }_{\psi }\mathrm{\mid }\det DG\mathrm{\mid }$. At a face point use the local extension $\widehat{G}$ of [F2]; on its negative side both zero-extended coefficients vanish, and on its positive and zero sides the displayed coefficient relation holds. The compact supports lie inside the extension neighborhoods, so the target zero extension has compact support in $\widehat{G}(O)$ and is Riemann integrable by [F1]. The source zero extension is Riemann integrable for the same reason. Apply [F3] to $\widehat{G}:O\to \widehat{G}(O)$ and multiply by the chart signs. This gives equality of the two signed chart integrals of this summand. At an interior point the identical argument uses $G$ directly. The genuine face causes no extra integral: [F1] makes it content zero, and the side-preserving extension ensures the zero extensions match on both sides. Sum the finitely many summands by [F5]. When $n=0$, two connected charts containing the support name the same point, and both signed evaluations agree. Hence $I_{\varphi }(\zeta )=I_{\psi }(\zeta )$ in every dimension, without invoking the countable-choice-qualified general chart-independence theorem.

Let $({\chi }_i,{\varphi }_i)$ and $({\tau }_j,{\psi }_j)$ be two finite localizations near $\mathrm{supp}\omega$. Because both sums equal one there and $\omega =0$ elsewhere, ${\chi }_i\omega ={\sum }_j{\chi }_i{\tau }_j\omega$ and ${\tau }_j\omega ={\sum }_i{\chi }_i{\tau }_j\omega$ globally. Every product form has compact support inside both corresponding chart domains. Step 4.1 compares its two chart integrals, and finite linearity [F5] therefore identifies both localization sums with ${\sum }_{i,j}I({\chi }_i{\tau }_j\omega )$. This proves independence. A form already supported in one chart has its chart integral as its ${\mathcal{I}}_M$ value by taking the construction of steps 1.1 and 2.1 inside that chart.

The union of two compact supports is compact. Choose one finite localization near that union; finite Riemann linearity gives linearity of ${\mathcal{I}}_M$, and step 5.1 removes dependence on that choice. If an open set $V\subseteq M$ contains the support, take every chart-and-bump tuple in step 1.1 inside $V$; the same finite chart sum computes ${\mathcal{I}}_V(\omega {\mathrm{\mid }}_V)$ and ${\mathcal{I}}_M(\omega )$. For $n=0$, the formula is the finite signed sum over the support, including empty support.

Let $F:M\to N$ be an orientation-preserving diffeomorphism and $\omega \in {\mathrm{\Omega }}_c^n(N)$. Its pullback has compact support $F^{-1}(\mathrm{supp}\omega )$, the continuous image of the target compact support under $F^{-1}$. Take a finite localization $({\chi }_i,{\varphi }_i)$ of $\omega$ on $N$. The functions ${\chi }_i\circ F$ and charts ${\varphi }_i\circ F$ localize $F^{\ast }\omega$ on $M$; their supports are compact and their sums equal one near its support. In corresponding coordinates the localized coefficients are identical by pullback functoriality, and the orientation signs agree. Step 5.1 therefore gives equal integrals. Under global reversal the coordinate signs are opposite, so the result is negated. If the sign varies, it is locally constant; compact support meets only finitely many open components by a finite-subcover argument, and the equality applies on each component. At $n=0$ this is a finite bijection of signed point values.

If $\omega$ is nonnegative on the chosen orientation ray, every localized signed chart coefficient of ${\chi }_i\omega$ is nonnegative, so [F5] makes every summand nonnegative. If $\omega \mathrm{\ne }0$, choose $p$ where it is positive. Since ${\sum }_i{\chi }_i(p)=1$, at least one ${\chi }_i(p)>0$. That summand's signed coefficient is positive at the coordinate point and hence at least $c>0$ on a sufficiently small closed interior rectangle $Q$ of positive volume; if $p$ is on the genuine face, place $Q$ just inside its positive side. Coordinate-slice additivity separates $Q$ from a surrounding bounding rectangle, and monotonicity bounds the integral on $Q$ below by $c\mathrm{vol}(Q)>0$ and every complementary piece below by zero. Thus this chart integral is strictly positive while all other chart terms are nonnegative. For $n=0$, the nonzero nonnegative signed point value supplies the strict inequality directly.

Finally assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and let $({\rho }_a)$ be a subordinate locally finite chart partition used by the published global definition. Only finitely many ${\rho }_a\omega$ are nonzero by [F6], and each has compact support in its chart. Insert a finite localization $({\chi }_i)$ into each such form. The product chart integrals agree by step 4.1, so finite distributivity gives ${\sum }_{a}I({\rho }_a\omega )={\sum }_{a,i}I({\rho }_a{\chi }_i\omega )={\sum }_{i}I({\chi }_i\omega )={\mathcal{I}}_M(\omega )$. Thus the choice-free finite value agrees with the existing global integral whenever that partition construction is available. No step of the preceding construction uses ${\mathrm{A}\mathrm{C}}_{\omega }$.