Landweber exact formal group laws and smooth cohomology theories

Bunke, Ulrich¹ ; Schick, Thomas² ; Schröder, Ingo² ; Wiethaup, Moritz²

¹NWF I - Mathematik, Universität Regensburg, 93040 Regensburg, Germany
²Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr 3, 37073 Göttingen, Germany

Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1751-1790

Résumé

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 W → M with a fixed U–structure and U–connection on the (stable) normal bundle of W → M.

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) ⊗MU∗R defines a multiplicative smooth extension of R(M) := MU(M) ⊗MU∗R 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.

Détail

DOI : 10.2140/agt.2009.9.1751

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

Bibliographie

[1] J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Math., Univ. of Chicago Press (1974)

[2] J L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Math. 107, Birkhäuser (1993)

[3] U Bunke, M Kreck, T Schick, A geometric description of smooth cohomology, to appear in Ann. Math. Blaise Pascal

[4] 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

[5] U Bunke, T Schick, Uniqueness of the extensions of generalized cohomology theories, submitted

[6] U Bunke, T Schick, On the topology of $T$–duality, Rev. Math. Phys. 17 (2005) 77

[7] J Cheeger, J Simons, Differential characters and geometric invariants, from: "Geometry and topology (College Park, Md., 1983/84)" (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 50

[8] J L Dupont, R Ljungmann, Integration of simplicial forms and Deligne cohomology, Math. Scand. 97 (2005) 11

[9] J Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43

[10] D S Freed, Dirac charge quantization and generalized differential cohomology, from: "Surveys in differential geometry" (editor S T Yau), Surv. Differ. Geom. VII, Int. Press (2000) 129

[11] D S Freed, M Hopkins, On Ramond–Ramond fields and $K$–theory, J. High Energy Phys. (2000) 14

[12] P Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997) 155

[13] F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Math. E20, Friedr. Vieweg Sohn (1992)

[14] M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M–theory, J. Differential Geom. 70 (2005) 329

[15] L Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Math., Springer (2003)

[16] P S Landweber, Homological properties of comodules over $\mathrm{MU}_{*}(\mathrm{MU})$ and $\mathrm{BP}_{*}(\mathrm{BP})$, Amer. J. Math. 98 (1976) 591

[17] P S Landweber, D C Ravenel, R E Stong, Periodic cohomology theories defined by elliptic curves, from: "The Čech centennial (Boston, MA, 1993)" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 317

[18] J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press (1974)

[19] D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293

[20] G De Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grund. der Math. Wissenschaften 266, Springer (1984)

[21] J Simons, D Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008) 45

[22] R E Stong, Notes on cobordism theory, Math. notes, Princeton Univ. Press (1968)

[23] R M Switzer, Algebraic topology—homotopy and homology, Classics in Math., Springer (2002)