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

Abstract

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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 S H * ( X ) . In this description smooth manifolds in Quillen’s description are replaced by so-called stratifolds, which are certain stratified spaces. The cohomology theory S H * ( 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.

How to cite

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Bunke, 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

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  2. Ulrich Bunke, Thomas Schick, Smooth K-Theory, (2009) Zbl1202.19007
  3. Ulrich Bunke, Thomas Schick, Uniqueness of smooth extensions of generalized cohomology theories, (2010) Zbl1252.55002
  4. 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
  5. 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
  6. Johan L. Dupont, Rune Ljungmann, Integration of simplicial forms and Deligne cohomology, Math. Scand. 97 (2005), 11-39 Zbl1101.14024MR2179587
  7. Reese Harvey, Blaine Lawson, From sparks to grundles—differential characters, Comm. Anal. Geom. 14 (2006), 25-58 Zbl1116.53048MR2230569
  8. M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), 329-452 Zbl1116.58018MR2192936
  9. Lars Hörmander, The analysis of linear partial differential operators. I, (2003), Springer-Verlag, Berlin Zbl1028.35001MR1996773
  10. Manuel Köhler, Integration in glatter Kohomologie, (2007) 
  11. Matthias Kreck, Differential algebraic topology, (2007) 
  12. Matthias Kreck, Wilhelm Singhoff, Homology and cohomology theories on manifolds, (2010) Zbl1226.57035
  13. James Simons, Dennis Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008), 45-56 Zbl1163.57020MR2365651

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