Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Characteristic classes in $ TMF$ of level $ \Gamma_1(3)$


Author: Gerd Laures
Journal: Trans. Amer. Math. Soc. 368 (2016), 7339-7357
MSC (2010): Primary 55N34, 55R40; Secondary 55P50, 22E66
DOI: https://doi.org/10.1090/tran/6575
Published electronically: November 6, 2015
MathSciNet review: 3471093
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ TMF_1(n)$ be the spectrum of topological modular forms
equipped with a $ \Gamma _1(n)$-structure. We compute the $ K(2)$-local $ TMF_1(3)$-cohomology of $ B{\mathit String}$ and $ B{\mathit Spin}$: both are power series rings freely generated by classes that we explicitly construct and which generalize the classical Pontryagin classes. As a first application of this computation, we show how to construct $ TMF(3n)$-cohomology classes from stable positive energy representations of the loop groups $ L{\mathit Spin}$.


References [Enhancements On Off] (What's this?)

  • [ABP66] D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, $ {\rm SU}$-cobordism, $ {\rm KO}$-characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54-67. MR 0189043 (32 #6470)
  • [AHR] M. Ando, M. Hopkins, and C. Rezk, Multiplicative orientations of ko-theory and of the spectrum of topological modular forms, preprint, 2014.
  • [AHS01] M. Ando, M. J. Hopkins, and N. P. Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001), no. 3, 595-687. MR 1869850 (2002g:55009), https://doi.org/10.1007/s002220100175
  • [AS68] M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484-530. MR 0236950 (38 #5243)
  • [AS69] M. F. Atiyah and G. B. Segal, Equivariant $ K$-theory and completion, J. Differential Geometry 3 (1969), 1-18. MR 0259946 (41 #4575)
  • [Bak94] Andrew Baker, Elliptic genera of level $ N$ and elliptic cohomology, J. London Math. Soc. (2) 49 (1994), no. 3, 581-593. MR 1271552 (95e:55009), https://doi.org/10.1112/jlms/49.3.581
  • [Bou79] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257-281. MR 551009 (80m:55006), https://doi.org/10.1016/0040-9383(79)90018-1
  • [Bry90] Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990), no. 4, 461-480. MR 1071369 (91j:58151), https://doi.org/10.1016/0040-9383(90)90016-D
  • [DR73] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143-316 (French). MR 0337993 (49 #2762)
  • [Eis95] David Eisenbud, Commutative algebra, With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [Fra92] Jens Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992), 43-65. MR 1235295 (94h:55007), https://doi.org/10.1002/mana.19921580104
  • [Goe10] Paul G. Goerss, Topological modular forms [after Hopkins, Miller and Lurie], Séminaire Bourbaki. Volume 2008/2009. Exposés 997-1011, Astérisque 332 (2010), Exp. No. 1005, viii, 221-255. MR 2648680 (2011m:55003)
  • [HBJ92] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, With appendices by Nils-Peter Skoruppa and by Paul Baum, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. MR 1189136 (94d:57001)
  • [HL] Michael Hill and Tyler Lawson, Topological modular forms with level structure, arXiv:1312.7394, 2013.
  • [HR95] Mark A. Hovey and Douglas C. Ravenel, The $ 7$-connected cobordism ring at $ p=3$, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3473-3502. MR 1297530 (95m:55008), https://doi.org/10.2307/2155020
  • [HS99a] Mark Hovey and Hal Sadofsky, Invertible spectra in the $ E(n)$-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), no. 1, 284-302. MR 1722151 (2000h:55017), https://doi.org/10.1112/S0024610799007784
  • [HS99b] 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), https://doi.org/10.1090/memo/0666
  • [JW85] David Copeland Johnson and W. Stephen Wilson, The Brown-Peterson homology of elementary $ p$-groups, Amer. J. Math. 107 (1985), no. 2, 427-453. MR 784291 (86j:55008), https://doi.org/10.2307/2374422
  • [KL02] Nitu Kitchloo and Gerd Laures, Real structures and Morava $ K$-theories, $ K$-Theory 25 (2002), no. 3, 201-214. MR 1909866 (2003i:55005), https://doi.org/10.1023/A:1015683917463
  • [KLW04a] Nitu Kitchloo, Gerd Laures, and W. Stephen Wilson, The Morava $ K$-theory of spaces related to $ BO$, Adv. Math. 189 (2004), no. 1, 192-236. MR 2093483 (2005k:55002), https://doi.org/10.1016/j.aim.2003.10.008
  • [KLW04b] Nitu Kitchloo, Gerd Laures, and W. Stephen Wilson, Splittings of bicommutative Hopf algebras, J. Pure Appl. Algebra 194 (2004), no. 1-2, 159-168. MR 2086079 (2005e:55019), https://doi.org/10.1016/j.jpaa.2004.04.002
  • [KM85] Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569 (86i:11024)
  • [KW88] Victor G. Kac and Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 70 (1988), no. 2, 156-236. MR 954660 (89h:17036), https://doi.org/10.1016/0001-8708(88)90055-2
  • [Lau99] Gerd Laures, The topological $ q$-expansion principle, Topology 38 (1999), no. 2, 387-425. MR 1660325 (2000c:55009), https://doi.org/10.1016/S0040-9383(98)00019-6
  • [Lau04] Gerd Laures, $ K(1)$-local topological modular forms, Invent. Math. 157 (2004), no. 2, 371-403. MR 2076927 (2005h:55003), https://doi.org/10.1007/s00222-003-0355-y
  • [LN12] Tyler Lawson and Niko Naumann, Commutativity conditions for truncated Brown-Peterson spectra of height 2, J. Topol. 5 (2012), no. 1, 137-168. MR 2897051, https://doi.org/10.1112/jtopol/jtr030
  • [LO] Gerd Laures and Martin Olbermann, $ tmf_0(3)$ characteristic classes of string bundles, arXiv:1403.7301, 2014.
  • [Mat] Akhil Mathew, The homology of $ tmf$, arXiv:1305.6100, 2013.
  • [Mil89] Haynes Miller, The elliptic character and the Witten genus, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 281-289. MR 1022688 (90i:55005), https://doi.org/10.1090/conm/096/1022688
  • [MR09] Mark Mahowald and Charles Rezk, Topological modular forms of level 3, Pure Appl. Math. Q. 5 (2009), no. 2, Special Issue: In honor of Friedrich Hirzebruch, 853-872. MR 2508904 (2010g:55010), https://doi.org/10.4310/PAMQ.2009.v5.n2.a9
  • [PS86] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. MR 900587 (88i:22049)
  • [RW80] Douglas C. Ravenel and W. Stephen Wilson, The Morava $ K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691-748. MR 584466 (81i:55005), https://doi.org/10.2307/2374093
  • [RWY98] Douglas C. Ravenel, W. Stephen Wilson, and Nobuaki Yagita, Brown-Peterson cohomology from Morava $ K$-theory, $ K$-Theory 15 (1998), no. 2, 147-199. MR 1648284 (2000d:55012), https://doi.org/10.1023/A:1007776725714
  • [Su07] Hsin-hao Su, The E(1,2) cohomology of the Eilenberg-MacLane space K(Z ,3), ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)-The Johns Hopkins University. MR 2709571
  • [Wit88] Edward Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 161-181. MR 970288, https://doi.org/10.1007/BFb0078045

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 55N34, 55R40, 55P50, 22E66

Retrieve articles in all journals with MSC (2010): 55N34, 55R40, 55P50, 22E66


Additional Information

Gerd Laures
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, NA1/66, D-44780 Bochum, Germany

DOI: https://doi.org/10.1090/tran/6575
Received by editor(s): March 26, 2014
Received by editor(s) in revised form: August 20, 2014, September 5, 2014, September 23, 2014, and September 30, 2014
Published electronically: November 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society