×

zbMATH — the first resource for mathematics

Landweber exact formal group laws and smooth cohomology theories. (English) Zbl 1181.55006
A smooth extension of a generalized cohomology theory \(h^*(\;)\) is a functor \(\widehat{h}^*(\;)\) from smooth compact manifolds to graded groups that is related to \(h^*(\;)\) in the way that deRham cohomology is related to ordinary integral cohomology. The paper under review gives a precise definition of this idea and shows how to construct such an extension in the case \(h^*(\;) = MU^*(\;)\), complex cobordism, using cycles which are essentially proper maps \(W\rightarrow M\) of smooth manifolds with a fixed \(U\)-structure and \(U\)-connection on the (stable) normal bundle. The authors establish an extension of the Landweber exact functor theorem showing that \(\widehat{MU}^* (\;)\otimes_{MU^*}{R}\) will be a smooth cohomology theory if the \(MU_*\) module \(R\) satisfies the exactness conditions in Landweber’s theorem. They also give conditions under which the extensions obtained are multiplicative and develop a smooth version of \(MU\)-orientation.

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
57R19 Algebraic topology on manifolds and differential topology
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Math., Univ. of Chicago Press (1974) · Zbl 0309.55016
[2] J L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Math. 107, Birkhäuser (1993) · Zbl 0823.55002
[3] U Bunke, M Kreck, T Schick, A geometric description of smooth cohomology, to appear in Ann. Math. Blaise Pascal · Zbl 1200.55007
[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 · Zbl 1202.19007
[5] U Bunke, T Schick, Uniqueness of the extensions of generalized cohomology theories, submitted · Zbl 1252.55002
[6] U Bunke, T Schick, On the topology of \(T\)-duality, Rev. Math. Phys. 17 (2005) 77 · Zbl 1148.55009
[7] J Cheeger, J Simons, Differential characters and geometric invariants (editors J Alexander, J Harer), Lecture Notes in Math. 1167, Springer (1985) 50 · Zbl 0621.57010
[8] J L Dupont, R Ljungmann, Integration of simplicial forms and Deligne cohomology, Math. Scand. 97 (2005) 11 · Zbl 1101.14024
[9] J Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43 · Zbl 0777.55003
[10] D S Freed, Dirac charge quantization and generalized differential cohomology (editor S T Yau), Surv. Differ. Geom. VII, Int. Press (2000) 129 · Zbl 1058.81058
[11] D S Freed, M Hopkins, On Ramond-Ramond fields and \(K\)-theory, J. High Energy Phys. (2000) 14 · Zbl 0990.81624
[12] P Gajer, Geometry of Deligne cohomology, Invent. Math. 127 (1997) 155 · Zbl 0873.14025
[13] F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Math. E20, Friedr. Vieweg & Sohn (1992) · Zbl 0767.57014
[14] M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005) 329 · Zbl 1116.58018
[15] L Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Math., Springer (2003) · Zbl 1028.35001
[16] P S Landweber, Homological properties of comodules over \(\mathrm{MU}_*(\mathrm{MU})\) and \(\mathrm{BP}_*(\mathrm{BP})\), Amer. J. Math. 98 (1976) 591 · Zbl 0355.55007
[17] P S Landweber, D C Ravenel, R E Stong, Periodic cohomology theories defined by elliptic curves (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 317 · Zbl 0920.55005
[18] J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press (1974) · Zbl 0298.57008
[19] D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293 · Zbl 0199.26705
[20] G de Rham, Differentiable manifolds. Forms, currents, harmonic forms, Grund. der Math. Wissenschaften 266, Springer (1984) · Zbl 0534.58003
[21] J Simons, D Sullivan, Axiomatic characterization of ordinary differential cohomology, J. Topol. 1 (2008) 45 · Zbl 1163.57020
[22] R E Stong, Notes on cobordism theory, Math. notes, Princeton Univ. Press (1968) · Zbl 0181.26604
[23] R M Switzer, Algebraic topology-homotopy and homology, Classics in Math., Springer (2002) · Zbl 1003.55002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.