Differential forms on diffeological spaces and diffeological gluing, I
Ekaterina Pervova
TL;DR
This work analyzes how diffeological differential forms behave under gluing along a smooth map $f:X_1\supset Y\to X_2$. The core result shows that $\Omega^m(X_1\cup_f X_2)$ is diffeomorphic to the space of compatible, $f$-invariant pairs $\Omega_f^m(X_1)\oplus_{comp}\Omega^m(X_2)$, via the pullback map $\pi^*$, with a well-defined induced form $\omega_1\cup_f\omega_2$ on the glued space. It also demonstrates that Souriau's de Rham complex on the glued space embeds as a subcomplex of the direct sum complex, and extends these ideas to iterated gluings and gluing along diffeomorphisms, supported by explicit examples and a treatment of corners. The results provide a precise framework for transferring and extending diffeological differential forms across glued diffeological spaces, clarifying both local compatibility and global extendibility properties. The methods have potential implications for constructing and analyzing smooth structures on spaces built by gluing non-manifold pieces.
Abstract
This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available, that of the vector space of all diffeological forms on a given space, and that of the pseudo-bundle of values of these forms. We describe the behavior of the former under a gluing of diffeological spaces.