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Quillen's work on formal group laws and complex cobordism theory

Published online by Cambridge University Press:  01 March 2013

Douglas C. Ravenel
Douglas C. Ravenel*
Affiliation:
Department of Mathematics, Rochester University, Rochester, NY, [email protected]
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Abstract

In 1969 Quillen discovered a deep connection between complex cobordism and formal group laws which he announced in [Qui69]. Algebraic topology has never been the same since. We will describe the content of [Qui69] and then discuss its impact on the field. This paper is a writeup of a talk on the same topic given at the Quillen Conference at MIT in October 2012. Slides for that talk are available on the author's home page.

Keywords

Formal group lawcomplex cobordismchromatic filtrationelliptic cohomologytopological modular forms55N2255N3455S2555T1514L0557R77
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 3: The Legacy of Daniel Quillen , June 2013 , pp. 493 - 506
Copyright
Copyright © ISOPP 2013 

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References

Ada58.Adams, J. F.. On the structure and applications of the Steenrod algebra. Comment. Math. Helv. 32:180–214, 1958.Google Scholar
Ada66.Adams, J. F.. On the groups J(X). IV. Topology 5:21–71, 1966.Google Scholar
Ada74.Adams, J. F.. Stable homotopy and generalised homology. University of Chicago Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics.Google Scholar
Ada92.Adams, J. Frank. The selected works of J. Frank Adams. Vol. II. Cambridge University Press, Cambridge, 1992. Edited and with an introduction and biographical data by May, J. P. and Thomas, C. B..Google Scholar
Bau08.Bauer, Tilman. Computation of the homotopy of the spectrum tmf. In Groups, homotopy and configuration spaces, Geom. Topol. Monogr. 13, pages 11–40. Geom. Topol. Publ., Coventry, 2008.Google Scholar
Beh.Behrens, Mark. Notes on the construction of tmf. To appear in proceedings of 2007 Talbot Workshop, and available on the author's home page.Google Scholar
BL10.Behrens, Mark and Lawson, Tyler. Topological automorphic forms. 2010.Google Scholar
Bot59.Bott, Raoul. The stable homotopy of the classical groups. Ann. of Math. (2) 70:313–337, 1959.Google Scholar
Bou79.Bousfield, A. K.. The localization of spectra with respect to homology. Topology 18(4):257–281, 1979.Google Scholar
BP66.Brown, E. H. and Peterson, F. P.. A spectrum whose Zp cohomology is the algebra of reduced p-th powers. Topology 5:149–154, 1966.Google Scholar
Car67.Cartier, Pierre. Modules associés à un groupe formel commutatif. Courbes typiques. C. R. Acad. Sci. Paris Sér. A-B 265:A129–A132, 1967.Google Scholar
CF66.Conner, P. E. and Floyd, E. E.. The relation of cobordism to K-theories. Lecture Notes in Mathematics 28, Springer-Verlag, Berlin, 1966.Google Scholar
DHS88.Devinatz, Ethan S., Hopkins, Michael J., and Smith, Jeffrey H.. Nilpotence and stable homotopy theory. I. Ann. of Math. (2) 128 (2):207–241, 1988.Google Scholar
Goe04.Goerss, Paul G.. (Pre-)sheaves of ring spectra over the moduli stack of formal group laws. In Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem. 131, pages 101–131. Kluwer Acad. Publ., Dordrecht, 2004.Google Scholar
Goe10.Goerss, Paul G.. Topological modular forms [after Hopkins, Miller and Lurie]. Astérisque 332: Exp. No. 1005, viii, 221–255, 2010. Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011.Google Scholar
Haz78.Hazewinkel, Michiel. Formal groups and applications, Pure and Applied Mathematics 78, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
Hen.Henriques, André. The homotopy groups of tmf and its localizations. To appear in proceedings of 2007 Talbot Workshop, and available on the author's home page.Google Scholar
HM.Hopkins, Michael J. and Mahowald, M. A.. From elliptic curves to homotopy theory. Preprint in Hopf archive at http://hopf.math.purdue.edu/Hopkins-Mahowald/eo2homotopy.Google Scholar
JW75.Johnson, David Copeland and Wilson, W. Stephen. BP operations and Morava's extraordinary K-theories. Math. Z. 144(1):55–75, 1975.Google Scholar
Lan88.Landweber, Peter S.. Elliptic cohomology and modular forms. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math. 1326, pages 55–68. Springer, Berlin, 1988.Google Scholar
Laz55.Lazard, Michel. Sur les groupes de Lie formels à un paramètre. Bull. Soc. Math. France 83:251–274, 1955.Google Scholar
LRS95.Landweber, Peter S., Ravenel, Douglas C., and Stong, Robert E.. Periodic cohomology theories defined by elliptic curves. In The Čech centennial (Boston, MA, 1993), Contemp. Math. 181, pages 317–337, Providence, RI, 1995. Amer. Math. Soc.Google Scholar
Lur09.Lurie, J.. A survey of elliptic cohomology. In Algebraic topology, Abel Symp. 4, pages 219–277. Springer, Berlin, 2009.CrossRefGoogle Scholar
Mil60.Milnor, J.. On the cobordism ring Ω* and a complex analogue. I. Amer. J. Math. 82:505–521, 1960.CrossRefGoogle Scholar
Mor85.Morava, Jack. Noetherian localisations of categories of cobordism comodules. Ann. of Math. (2) 121(1):1–39, 1985.Google Scholar
MR77.Miller, Haynes R. and Ravenel, Douglas C.. Morava stabilizer algebras and the localization of Novikov's E 2-term. Duke Math. J. 44(2):433–447, 1977.Google Scholar
MR07.Miller, Haynes R. and Ravenel, Douglas C., editors. Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, London Mathematical Society Lecture Note Series 342, Cambridge, 2007. Cambridge University Press.Google Scholar
MRW77.Miller, Haynes R., Ravenel, Douglas C., and Wilson, W. Stephen. Periodic phenomena in the Adams-Novikov spectral sequence. Ann. Math. (2) 106(3):469–516, 1977.Google Scholar
Nas.Nassau, Christian. Cohomology charts.Google Scholar
Nov67.Novikov, S. P.. Methods of algebraic topology from the point of view of cobordism theory. Izv. Akad. Nauk SSSR Ser. Mat. 31:855–951, 1967.Google Scholar
Och87.Ochanine, Serge. Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology 26(2):143–151, 1987.Google Scholar
Qui69.Quillen, Daniel. On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75:1293–1298, 1969.Google Scholar
Rav76.Ravenel, Douglas C.. The structure of Morava stabilizer algebras. Inventiones Math. 37:109–120, 1976.Google Scholar
Rav77.Ravenel, Douglas C.. The cohomology of Morava stabilizer algebras. Mathematische Zeitschrift 152:287–297, 1977.Google Scholar
Rav82.Ravenel, Douglas C.. Morava K-theories and finite groups. In Gitler, S., editor, Symposium on Algebraic Topology in Honor of José Adem, Contemporary Mathematics, pages 289–292, Providence, Rhode Island, 1982. American Mathematical Society.Google Scholar
Rav84.Ravenel, Douglas C.. Localization with respect to certain periodic homology theories. Amer. J. Math. 106(2):351–414, 1984.Google Scholar
Rav86.Ravenel, Douglas C.. Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121. Academic Press Inc., Orlando, FL, 1986. Errata and second edition available online at author's home page.Google Scholar
Rav92.Ravenel, Douglas C.. Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128. Princeton University Press, Princeton, NJ, 1992. Appendix C by Smith, Jeff.Google Scholar
RW80.Ravenel, Douglas C. and Wilson, W. Stephen. The Morava K-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture. Amer. J. Math. 102(4):691–748, 1980.Google Scholar
Smi77.Smith, Larry. On realizing complex bordism modules. IV. Applications to the stable homotopy groups of spheres. Amer. J. Math. 99(2):418–436, 1977.Google Scholar
ST04.Stolz, Stephan and Teichner, Peter. What is an elliptic object? In Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 308, pages 247–343. Cambridge Univ. Press, Cambridge, 2004.Google Scholar
Tat68.Tate, John. Residues of differentials on curves. Ann. Sci. École Norm. Sup. (4) 1:149–159, 1968.Google Scholar
Tod62.Toda, Hirosi. Composition methods in homotopy groups of spheres. Annals of Mathematics Studies 49. Princeton University Press, Princeton, N.J., 1962.Google Scholar
Tod71.Toda, Hirosi. On spectra realizing exterior parts of the Steenrod algebra. Topology 10:53–65, 1971.Google Scholar
Whi42.Whitehead, George W.. On the homotopy groups of spheres and rotation groups. Ann. of Math. (2) 43:634–640, 1942.Google Scholar
Wil84.Wilson, W. Stephen. The Hopf ring for Morava K-theory. Publ. Res. Inst. Math. Sci. 20(5):1025–1036, 1984.Google Scholar
Wit87.Witten, Edward. Elliptic genera and quantum field theory. Comm. Math. Phys. 109(4):525–536, 1987.Google Scholar
Wit88.Witten, Edward. The index of the Dirac operator in loop space. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math. 1326, pages 161–181, Berlin, 1988. Springer.Google Scholar
Yag80.Yagita, Nobuaki. On the Steenrod algebra of Morava K-theory. J. London Math. Soc. (2) 22(3):423–438, 1980.Google Scholar