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ON THE DOUBLE TRANSFER AND THE f-INVARIANT

Published online by Cambridge University Press:  30 March 2012

GEOFFREY POWELL
GEOFFREY POWELL*
Affiliation:
Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France e-mail: powell@math.univ-paris13.fr
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Abstract

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The purpose of this paper is to investigate the algebraic double S1-transfer, in particular the classes in the two-line of the Adams–Novikov spectral sequence which are the image of comodule primitives of the MU-homology of ℂP∞ × ℂP∞ via the algebraic double transfer. These classes are analysed by two related approaches: the first, p-locally for p ≥ 3, by using the morphism induced in MU-homology by the chromatic factorisation of the double transfer map together with the f′-invariant of Behrens (for p ≥ 5) (M. Behrens, Congruences between modular forms given by the divided β-family in homotopy theory, Geom. Topol.13(1) (2009), 319–357). The second approach (after inverting 6) uses the algebraic double transfer and the f-invariant of Laures (G. Laures, The topological q-expansion principle, Topology38(2) (1999), 387–425).

Keywords

Primary 55R12Secondary 55N3455P42
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 547 - 577
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Baker, A., On the detection of some elements in the image of the double transfer using K(2)-theory, Math. Z. 197 (3) (1988), 439–454. MR926851 (89c:55019)CrossRefGoogle Scholar
2.Baker, A., Carlisle, D., Gray, B., Hilditch, S., Ray, N. and Wood, R., On the iterated complex transfer, Math. Z. 199 (2) (1988), 191–207. MR958648 (90d:55021)CrossRefGoogle Scholar
3.Baker, A., Clarke, F., Ray, N. and Schwartz, L., On the Kummer congruences and the stable homotopy of BU, Trans. Amer. Math. Soc. 316 (2) (1989), 385–432. MR942424 (90c:55003)Google Scholar
4.Behrens, M., Congruences between modular forms given by the divided β family in homotopy theory, Geom. Topol. 13 (1) (2009), 319–357. MR2469520 (2009i:55016)CrossRefGoogle Scholar
5.Behrens, M. and Laures, G., β-family congruences and the f-invariant, new topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr. 16 (2009), 9–29. (Geom. Topol. Publ., Coventry) MR2544384CrossRefGoogle Scholar
6.Hovey, M. and Strickland, N., Comodules and Landweber exact homology theories, Adv. Math. 192 (2) (2005), 427–456. MR2128706 (2006e:55007)CrossRefGoogle Scholar
7.Imaoka, M., Double transfers at the prime 2, Sūrikaisekikenkyūsho Kōkyūroku 838 (1993), 1–8. (Developments and prospects in algebraic topology (Japanese), Kyoto, Japan). MR1289907Google Scholar
8.Imaoka, M., Factorization of double transfer maps, Osaka J. Math. 30 (4) (1993), 759–769. MR1250782 (95d:55015)Google Scholar
9.Katz, N. M., Higher congruences between modular forms, Ann. Math. 101 (2) (1975), 332–367. MR0417059 (54 #5120)CrossRefGoogle Scholar
10.Knapp, K., Some applications of K-theory to framed bordism, e-invariant and the transfer, mimeographed notes (Habilitationschrift, Universität Bonn, Germany, 1979).Google Scholar
11.Knapp, K., Introduction to nonconnective Im(J)-theory, in Handbook of algebraic topology (James, I. M., Editor) (North-Holland, Amsterdam, 1995), 425–461. MR1361896 (97a:55016)CrossRefGoogle Scholar
12.Laures, G., The topological q-expansion principle, Topology 38 (2) (1999), 387–425. MR1660325 (2000c:55009)CrossRefGoogle Scholar
13.Miller, H., Universal Bernoulli numbers and the S 1-transfer, in Current trends in algebraic topology, Part 2 (London, Ont., 1981), CMS Conference Proceedings, vol. 2, American Mathematical Society, Providence, RI, 1982, pp. 437–449. MR686158 (85b:55029)Google Scholar
14.Miller, H., The elliptic character and the Witten genus, Algebraic topology (Proceedings of the International Conference, Evanston, IL, 1988), Contemp. Math. 96 (1989), 281–289 (American Mathematical Society, Providence, RI, 1989). MR1022688 (90i:55005)CrossRefGoogle Scholar
15.Miller, H. R., Ravenel, D. C. and Wilson, W. S., Periodic phenomena in the Adams-Novikov spectral sequence, Ann. Math. (2) 106 (3) (1977), 469–516. MR0458423 (56 #16626)CrossRefGoogle Scholar
16.Ravenel, D. C., Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (2) (1984), 351–414. MR737778 (85k:55009)CrossRefGoogle Scholar
17.Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres, in Pure and applied mathematics (Ravenel, D. C., Editor), vol. 121 (Academic Press, Orlando, FL, 1986). MR860042 (87j:55003), xx+413.Google Scholar
18.Segal, D. M., The cooperations of MU* (CP ∞) and MU* (HP ∞) and the primitive generators, J. Pure Appl. Algebra 14 (3) (1979), 315–322. MR533431 (80j:55006)CrossRefGoogle Scholar
19.Switzer, R. M., Algebraic topology—homotopy and homology, in Classics in mathematics (Springer-Verlag, Berlin, Germany, 2002), Reprint of the 1975 original Springer, New York; MR0385836 (52 #6695). 1886843Google Scholar