Čech-de Rham bicomplex in diffeology. (English) Zbl 1537.55011
This paper introduces Čech cohomology for diffeological spaces, which are adapted to their specificity. A good approach in diffeology for the Čech cohomology is to be defined as the Hochschild cohomology of a gauge monoid, associated with the specific generating family of round plots. The computation in the case of the irrational torus shows that this Čech cohomology achieves its purpose, even in this extreme case.
The author connects this Čech cohomology with the de Rham cohomology through a double complex as in [A. Weil, Comment. Math. Helv. 26, 119–145 (1952; Zbl 0047.16702)]. This complex together with its associated two filtrations and the spectral sequence is described. Contrarily to the case of smooth manifolds, the spectral sequence does not split in general, containing the obstruction to the de Rham theorem in diffeology. This obstruction is identified with the middle term in the low-degree sequence of the spectral sequence, which gives to it a full-fledged geometric status. The geometric nature of the higher obstructions of the de Rham theorem remains to be challenged.
Editorial remark: As the author states, this is a translation and revision of the unpublished preprint from 1988 [“Une cohomologie de Čech pour les espaces différentiables et sa relation à la cohomologie de De Rham”, Preprint, http://math.huji.ac.il/~piz/documents/UCDCPLEDESRALCDDR.pdf.
The author connects this Čech cohomology with the de Rham cohomology through a double complex as in [A. Weil, Comment. Math. Helv. 26, 119–145 (1952; Zbl 0047.16702)]. This complex together with its associated two filtrations and the spectral sequence is described. Contrarily to the case of smooth manifolds, the spectral sequence does not split in general, containing the obstruction to the de Rham theorem in diffeology. This obstruction is identified with the middle term in the low-degree sequence of the spectral sequence, which gives to it a full-fledged geometric status. The geometric nature of the higher obstructions of the de Rham theorem remains to be challenged.
Editorial remark: As the author states, this is a translation and revision of the unpublished preprint from 1988 [“Une cohomologie de Čech pour les espaces différentiables et sa relation à la cohomologie de De Rham”, Preprint, http://math.huji.ac.il/~piz/documents/UCDCPLEDESRALCDDR.pdf.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 55N99 | Homology and cohomology theories in algebraic topology |
| 58A12 | de Rham theory in global analysis |
| 58A40 | Differential spaces |
Citations:
Zbl 0047.16702References:
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