The angular period and the obstruction to bounding

Countable choice is The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); it is used for the global Stokes and embedded-submanifold integration route in [F1].

A nonzero period obstructs exactness and bounding: Under ${\mathrm{A}\mathrm{C}}_{\omega }$, if $S\hookrightarrow M$ is an oriented compact boundaryless embedded $k$-submanifold, $k\ge 1$, and $\omega$ is a closed smooth $k$-form on $M$ with ${\int }_S\omega \mathrm{\ne }0$, then $\omega$ is not exact on $M$ and $S$ cannot be the induced oriented boundary of a compact embedded $(k+1)$-submanifold of $M$.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.