# A geometric description of differential cohomology

Ulrich Bunke^{[1]}; Matthias Kreck^{[2]}; Thomas Schick^{[3]}

- [1] NWF I - Mathematik Universität Regensburg 93040 Regensburg Deutschland
- [2] Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 45 D-53115 Bonn Germany
- [3] Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr. 3 37073 Göttingen Germany

Annales mathématiques Blaise Pascal (2010)

- Volume: 17, Issue: 1, page 1-16
- ISSN: 1259-1734

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topBunke, Ulrich, Kreck, Matthias, and Schick, Thomas. "A geometric description of differential cohomology." Annales mathématiques Blaise Pascal 17.1 (2010): 1-16. <http://eudml.org/doc/116350>.

@article{Bunke2010,

abstract = {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.},

affiliation = {NWF I - Mathematik Universität Regensburg 93040 Regensburg Deutschland; Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 45 D-53115 Bonn Germany; Mathematisches Institut Georg-August-Universität Göttingen Bunsenstr. 3 37073 Göttingen Germany},

author = {Bunke, Ulrich, Kreck, Matthias, Schick, Thomas},

journal = {Annales mathématiques Blaise Pascal},

keywords = {differential cohomology; smooth cohomology; geometric cycles; cobordism; stratifold; Stokes' theorem; integration along the fiber},

language = {eng},

month = {1},

number = {1},

pages = {1-16},

publisher = {Annales mathématiques Blaise Pascal},

title = {A geometric description of differential cohomology},

url = {http://eudml.org/doc/116350},

volume = {17},

year = {2010},

}

TY - JOUR

AU - Bunke, Ulrich

AU - Kreck, Matthias

AU - Schick, Thomas

TI - A geometric description of differential cohomology

JO - Annales mathématiques Blaise Pascal

DA - 2010/1//

PB - Annales mathématiques Blaise Pascal

VL - 17

IS - 1

SP - 1

EP - 16

AB - 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.

LA - eng

KW - differential cohomology; smooth cohomology; geometric cycles; cobordism; stratifold; Stokes' theorem; integration along the fiber

UR - http://eudml.org/doc/116350

ER -

## References

top- Nils Andreas Baas, On formal groups and singularities in complex cobordism theory, Math. Scand. 33 (1973), 303-313 (1974) Zbl0281.57028MR346825
- Ulrich Bunke, Thomas Schick, Smooth K-Theory, (2009) Zbl1202.19007
- Ulrich Bunke, Thomas Schick, Uniqueness of smooth extensions of generalized cohomology theories, (2010) Zbl1252.55002
- Ulrich Bunke, Thomas Schick, Ingo Schröder, Moritz Wiethaup, Landweber exact formal group laws and smooth cohomology theories, Algebr. Geom. Topol. 9 (2009), 1751-1790 Zbl1181.55006MR2550094
- Jeff Cheeger, James Simons, Differential characters and geometric invariants, Geometry and topology (College Park, Md., 1983/84) 1167 (1985), 50-80, Springer, Berlin Zbl0621.57010MR827262
- Johan L. Dupont, Rune Ljungmann, Integration of simplicial forms and Deligne cohomology, Math. Scand. 97 (2005), 11-39 Zbl1101.14024MR2179587
- Reese Harvey, Blaine Lawson, From sparks to grundles—differential characters, Comm. Anal. Geom. 14 (2006), 25-58 Zbl1116.53048MR2230569
- M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), 329-452 Zbl1116.58018MR2192936
- Lars Hörmander, The analysis of linear partial differential operators. I, (2003), Springer-Verlag, Berlin Zbl1028.35001MR1996773
- Manuel Köhler, Integration in glatter Kohomologie, (2007)
- Matthias Kreck, Differential algebraic topology, (2007)
- Matthias Kreck, Wilhelm Singhoff, Homology and cohomology theories on manifolds, (2010) Zbl1226.57035
- James Simons, Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008), 45-56 Zbl1163.57020MR2365651

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