A comparison between two de Rham complexes in diffeology

Katsuhiko Kuribayashi

Published 2020-02-17Version 1

There are two de Rham complexes in diffeology. The original one is due to Souriau and another one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the \v{C}ech--de Rham spectral sequence in diffeology. In particular, a comparison map enables us to conclude that the first singular de Rham cohomology for the irrational torus $T_\theta$ is isomorphic to the direct sum of the original one and the group of equivalence classes of flow bundles over $T_\theta$ with connection $1$-forms.