Algebraic & Geometric Topology

Landweber exact formal group laws and smooth cohomology theories

Ulrich Bunke, Thomas Schick, Ingo Schröder, and Moritz Wiethaup

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Abstract

The main aim of this paper is the construction of a smooth (sometimes called differential) extension MÛ of the cohomology theory complex cobordism MU, using cycles for MÛ(M) which are essentially proper maps WM with a fixed U–structure and U–connection on the (stable) normal bundle of WM.

Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties.

Moreover, we show that R̂(M):=MÛ(M)MUR defines a multiplicative smooth extension of R(M):=MU(M)MUR whenever R is a Landweber exact MU–module, by using the Landweber exact functor principle. An example for this construction is a new way to define a multiplicative smooth K–theory.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1751-1790.

Dates
Received: 24 September 2008
Revised: 15 July 2009
Accepted: 19 July 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797043

Digital Object Identifier
doi:10.2140/agt.2009.9.1751

Mathematical Reviews number (MathSciNet)
MR2550094

Zentralblatt MATH identifier
1181.55006

Subjects
Primary: 55N20: Generalized (extraordinary) homology and cohomology theories 57R19: Algebraic topology on manifolds

Keywords
differential cohomology generalized cohomology theory Landweber exact formal group law smooth cohomology bordism geometric construction of differential cohomology

Citation

Bunke, Ulrich; Schick, Thomas; Schröder, Ingo; Wiethaup, Moritz. Landweber exact formal group laws and smooth cohomology theories. Algebr. Geom. Topol. 9 (2009), no. 3, 1751--1790. doi:10.2140/agt.2009.9.1751. https://projecteuclid.org/euclid.agt/1513797043


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References

  • J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Math., Univ. of Chicago Press (1974)
  • J-L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Math. 107, Birkhäuser, Boston (1993)
  • U Bunke, M Kreck, T Schick, A geometric description of smooth cohomology, to appear in Ann. Math. Blaise Pascal
  • U Bunke, T Schick, Smooth $K$–theory, to appear in “From probability to geometry”, (X Ma, editor) Asterisque, Volume dedicated to J-M Bismut for his $60$–th birthday
  • U Bunke, T Schick, Uniqueness of the extensions of generalized cohomology theories, submitted
  • U Bunke, T Schick, On the topology of $T$–duality, Rev. Math. Phys. 17 (2005) 77–112
  • J Cheeger, J Simons, Differential characters and geometric invariants, from: “Geometry and topology (College Park, Md., 1983/84)”, (J Alexander, J Harer, editors), Lecture Notes in Math. 1167, Springer, Berlin (1985) 50–80
  • J L Dupont, R Ljungmann, Integration of simplicial forms and Deligne cohomology, Math. Scand. 97 (2005) 11–39
  • J Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43–65
  • D S Freed, Dirac charge quantization and generalized differential cohomology, from: “Surveys in differential geometry”, (S-T Yau, editor), Surv. Differ. Geom. VII, Int. Press, Somerville, MA (2000) 129–194
  • D S Freed, M Hopkins, On Ramond–Ramond fields and $K$–theory, J. High Energy Phys. (2000) Paper 44, 14
  • P Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997) 155–207
  • F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Math. E20, Friedr. Vieweg & Sohn, Braunschweig (1992) With appendices by N-P Skoruppa and by P Baum
  • M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M–theory, J. Differential Geom. 70 (2005) 329–452
  • L H örmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Math., Springer, Berlin (2003) Reprint of the second (1990) edition
  • P S Landweber, Homological properties of comodules over $\mathrm{MU}_{*}(\mathrm{MU})$ and $\mathrm{BP}_{*}(\mathrm{BP})$, Amer. J. Math. 98 (1976) 591–610
  • P S Landweber, D C Ravenel, R E Stong, Periodic cohomology theories defined by elliptic curves, from: “The Čech centennial (Boston, MA, 1993)”, (M Cenkl, H Miller, editors), Contemp. Math. 181, Amer. Math. Soc. (1995) 317–337
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press (1974)
  • D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293–1298
  • G de Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grund. der Math. Wissenschaften 266, Springer, Berlin (1984) Translated from the French by F R Smith, With an introduction by S S Chern
  • J Simons, D Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008) 45–56
  • R E Stong, Notes on cobordism theory, Math. notes, Princeton Univ. Press (1968)
  • R M Switzer, Algebraic topology–-homotopy and homology, Classics in Math., Springer, Berlin (2002) Reprint of the 1975 original