A geometric description of differential cohomology
Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, p. 1-16

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.

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.

DOI : https://doi.org/10.5802/ambp.276
Classification:  55N20,  57R19
Keywords: differential cohomology, smooth cohomology, geometric cycles, cobordism
     author = {Bunke, Ulrich and Kreck, Matthias and Schick, Thomas},
     title = {A geometric description of differential cohomology},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     number = {1},
     year = {2010},
     pages = {1-16},
     doi = {10.5802/ambp.276},
     mrnumber = {2674652},
     zbl = {1200.55007},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2010__17_1_1_0}
Bunke, Ulrich; Kreck, Matthias; Schick, Thomas. A geometric description of differential cohomology. Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, pp. 1-16. doi : 10.5802/ambp.276. http://www.numdam.org/item/AMBP_2010__17_1_1_0/

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

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

[4] Bunke, Ulrich; Schick, Thomas; Schröder, Ingo; Wiethaup, Moritz Landweber exact formal group laws and smooth cohomology theories, Algebr. Geom. Topol., Tome 9 (2009) no. 3, pp. 1751-1790 | Article | MR 2550094 | Zbl 1181.55006 | Zbl pre05610801

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

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

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

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

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

[10] Köhler, Manuel Integration in glatter Kohomologie, Georg-August-Universität Göttingen (2007) (Technical report)

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

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

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