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The units of a ring spectrum and a logarithmic cohomology operation

Author: Charles Rezk
Journal: J. Amer. Math. Soc. 19 (2006), 969-1014
MSC (2000): Primary 55N22; Secondary 55P43, 55S05, 55S25, 55P47, 55P60, 55N34, 11F25
Published electronically: February 8, 2006
MathSciNet review: 2219307
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Abstract: We construct a ``logarithmic'' cohomology operation on Morava $ E$-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring $ E^0(K)$ of a space $ K$. We obtain a formula for this map in terms of the action of Hecke operators on Morava $ E$-theory. Our formula is closely related to that for an Euler factor of the Hecke $ L$-function of an automorphic form.

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  • [AHS04] Matthew Ando, Michael J. Hopkins, and Neil P. Strickland, The sigma orientation is an $ H\sb \infty$ map, Amer. J. Math. 126 (2004), no. 2, 247-334. MR 2045503 (2005d:55009)
  • [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39:4129)
  • [And95] Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories $ E\sb n$, Duke Math. J. 79 (1995), no. 2, 423-485. MR 1344767 (97a:55006)
  • [AS71] M. F. Atiyah and G. B. Segal, Exponential isomorphisms for $ \lambda $-rings, Quart. J. Math. Oxford Ser. (2) 22 (1971), 371-378. MR 0291250 (45:344)
  • [BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of $ \gamma $-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Springer, Berlin, 1978, pp. 80-130. MR 0513569 (80e:55021)
  • [BH04] Martin Bendersky and John R. Hunton, On the coalgebraic ring and Bousfield-Kan spectral sequence for a Landweber exact spectrum, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 513-532. MR 2096616 (2005j:55010)
  • [BMMS86] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $ H\sb \infty $ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 0836132 (88e:55001)
  • [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257-281. MR 0551009 (80m:55006)
  • [Bou87] -, Uniqueness of infinite deloopings for $ {K}$-theoretic spaces, Pacific J. Math. 129 (1987), no. 1, 1-31. MR 0901254 (89g:55017)
  • [Bou01] -, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391-2426 (electronic). MR 1814075 (2001k:55030)
  • [Dol62] Albrecht Dold, Decomposition theorems for $ S(n)$-complexes, Ann. of Math. (2) 75 (1962), 8-16. MR 0137113 (25:569)
  • [EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole. MR 1417719 (97h:55006)
  • [GH04] P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. MR R2125040 (2006b:55010)
  • [GJ99] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612 (2001d:55012)
  • [Glu81] David Gluck, Idempotent formula for the Burnside algebra with applications to the $ p$-subgroup simplicial complex, Illinois J. Math. 25 (1981), no. 1, 63-67. MR 0602896 (82c:20005)
  • [HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553-594 (electronic). MR 1758754 (2001k:55015)
  • [Hod72] Luke Hodgkin, The $ K$-theory of some wellknown spaces. I. $ QS\sp{0}$, Topology 11 (1972), 371-375. MR 0331367 (48:9701)
  • [Hop02] M. J. Hopkins, Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 291-317. MR 1989190 (2004g:11032)
  • [HS99] Mark Hovey and Neil P. Strickland, Morava $ K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906 (99b:55017)
  • [HSS00] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149-208. MR 1695653 (2000h:55016)
  • [Kas94] Takuji Kashiwabara, Hopf rings and unstable operations, J. Pure Appl. Algebra 94 (1994), no. 2, 183-193. MR 1282839 (95h:55005)
  • [Kuh] Nicholas J. Kuhn, Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces, preprint.
  • [Kuh89] Nicholas J. Kuhn, Morava $ K$-theories and infinite loop spaces, Algebraic topology (Arcata, CA, 1986) (Berlin), Lecture Notes in Math., vol. 1370, Springer, 1989, pp. 243-257. MR 1000381 (90d:55014)
  • [LMSM86] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986, With contributions by J. E. McClure. MR 0866482 (88e:55002)
  • [LT66] Jonathan Lubin and John Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49-59. MR 0238854 (39:214)
  • [May72] J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin, 1972, Lectures Notes in Mathematics, Vol. 271. MR 0420610 (54:8623b)
  • [May77] J. Peter May, $ E\sb{\infty }$ ring spaces and $ E\sb{\infty }$ ring spectra, Springer-Verlag, Berlin, 1977, With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave, Lecture Notes in Mathematics, Vol. 577. MR 0494077 (58:13008)
  • [May82] J. P. May, Multiplicative infinite loop space theory, J. Pure Appl. Algebra 26 (1982), no. 1, 1-69. MR 0669843 (84c:55013)
  • [MM02] M. A. Mandell and J. P. May, Equivariant orthogonal spectra and $ S$-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 1922205 (2003i:55012)
  • [RR04] Birgit Richter and Alan Robinson, Gamma homology of group algebras and of polynomial algebras, Homotopy theory: Relations with algebraic geometry, group cohomology, and algebraic $ K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 453-461. MR 2066509 (2005k:55006)
  • [Shi71] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kanô Memorial Lectures, No. 1. MR 0314766 (47:3318)
  • [ST97] Neil P. Strickland and Paul R. Turner, Rational Morava $ E$-theory and $ DS\sp 0$, Topology 36 (1997), no. 1, 137-151. MR 1410468 (97g:55005)
  • [Str97] Neil P. Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997), no. 2, 161-208. MR 1473889 (98k:14065)
  • [Str98] N. P. Strickland, Morava $ E$-theory of symmetric groups, Topology 37 (1998), no. 4, 757-779. MR 1607736 (99e:55008)
  • [Woo79] Richard Woolfson, Hyper-$ \Gamma $-spaces and hyperspectra, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 118, 229-255. MR 0534835 (81b:55026)

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Additional Information

Charles Rezk
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61820

Received by editor(s): April 5, 2005
Published electronically: February 8, 2006
Additional Notes: This work was supported by the National Science Foundation under award DMS-0203936.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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