A geometric description of differential cohomology
[Une description géométrique de la cohomologie différentielle]
Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 1-16.

Nous donnons une définition géométrique de la cohomologie intégrale différentielle. Nous utilisons des cycles de cobordisme avec singularités, et des formes différentielles distributionnelles. Avec cette description, la construction de la multiplication et de l’intégration avec toutes les proprietés désirées est particulièrement simple.

In this paper we give a geometric cobordism description of differential integral cohomology. The main motivation to consider this model (for other models see [5, 6, 7, 8]) is that it allows for simple descriptions of both the cup product and the integration. In particular it is very easy to verify the compatibilty of these structures. We proceed in a similar way in the case of differential cobordism as constructed in [4]. There the starting point was Quillen’s cobordism description of singular cobordism groups for a differential manifold X. Here we use instead the similar description of integral cohomology from [11]. This cohomology theory is denoted by SH * (X). In this description smooth manifolds in Quillen’s description are replaced by so-called stratifolds, which are certain stratified spaces. The cohomology theory SH * (X) is naturally isomorphic to ordinary integral cohomology H * (X), thus we obtain a cobordism type definition of the differential extension of ordinary integral cohomology.

DOI : https://doi.org/10.5802/ambp.276
Classification : 55N20,  57R19
Mots clés : cohomologie différentielle, cycles géométriques, cobordisme
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     author = {Ulrich Bunke and Matthias Kreck and Thomas Schick},
     title = {A geometric description of differential cohomology},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {1--16},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     number = {1},
     year = {2010},
     doi = {10.5802/ambp.276},
     mrnumber = {2674652},
     zbl = {1200.55007},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.276/}
}
Ulrich Bunke; Matthias Kreck; Thomas Schick. A geometric description of differential cohomology. Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 1-16. doi : 10.5802/ambp.276. https://ambp.centre-mersenne.org/articles/10.5802/ambp.276/

[1] Nils Andreas Baas On formal groups and singularities in complex cobordism theory, Math. Scand., Volume 33 (1973), p. 303-313 (1974) | EuDML 166333 | MR 346825 | Zbl 0281.57028

[2] Ulrich Bunke; Thomas Schick Smooth K-Theory (2009) (arXiv:0707.0046, to appear in From Probability to Geometry. Volume dedicated to J.-M. Bismut for his 60th birthday (X. Ma, editor), Asterisque 327 & 328) | Zbl 1202.19007

[3] Ulrich Bunke; Thomas Schick Uniqueness of smooth extensions of generalized cohomology theories (2010) (arXiv.org:0901.4423, to appear in Journal of Topology) | Zbl 1252.55002

[4] Ulrich Bunke; Thomas Schick; Ingo Schröder; Moritz Wiethaup Landweber exact formal group laws and smooth cohomology theories, Algebr. Geom. Topol., Volume 9 (2009) no. 3, pp. 1751-1790 | Article | MR 2550094 | Zbl pre05610801

[5] Jeff Cheeger; James Simons Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) (Lecture Notes in Math.) Volume 1167, Springer, Berlin, 1985, pp. 50-80 | MR 827262 | Zbl 0621.57010

[6] Johan L. Dupont; Rune Ljungmann Integration of simplicial forms and Deligne cohomology, Math. Scand., Volume 97 (2005) no. 1, pp. 11-39 | MR 2179587 | Zbl 1101.14024

[7] Reese Harvey; Blaine Lawson From sparks to grundles—differential characters, Comm. Anal. Geom., Volume 14 (2006) no. 1, pp. 25-58 | MR 2230569 | Zbl 1116.53048

[8] M. J. Hopkins; I. M. Singer Quadratic functions in geometry, topology, and M-theory, J. Differential Geom., Volume 70 (2005) no. 3, pp. 329-452 | MR 2192936 | Zbl 1116.58018

[9] Lars Hörmander The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003 (Distribution theory and Fourier analysis, Reprint of the second (1990) edition) | MR 1996773 | Zbl 1028.35001

[10] Manuel Köhler Integration in glatter Kohomologie (2007) (Technical report)

[11] Matthias Kreck Differential algebraic topology (2007) (Preprint, available at http://www.hausdorff-research-institute.uni-bonn.de/kreck)

[12] Matthias Kreck; Wilhelm Singhoff Homology and cohomology theories on manifolds (2010) (to appear in Münster Journal of Mathematics)

[13] James Simons; Dennis Sullivan Axiomatic characterization of ordinary differential cohomology, J. Topol., Volume 1 (2008) no. 1, pp. 45-56 | Article | MR 2365651 | Zbl 1163.57020