Algebraic & Geometric Topology

The equivariant $J$–homomorphism for finite groups at certain primes

Christopher P French

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Abstract

Suppose G is a finite group and p a prime, such that none of the prime divisors of G are congruent to 1 modulo p. We prove an equivariant analogue of Adams’ result that J′=J′′. We use this to show that the G–connected cover of QGS0, when completed at p, splits up to homotopy as a product, where one of the factors of the splitting contains the image of the classical equivariant J–homomorphism on equivariant homotopy groups.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 4 (2009), 1885-1949.

Dates
Received: 7 May 2007
Revised: 17 July 2009
Accepted: 3 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797069

Digital Object Identifier
doi:10.2140/agt.2009.9.1885

Mathematical Reviews number (MathSciNet)
MR2550461

Zentralblatt MATH identifier
1177.19004

Subjects
Primary: 19L20: $J$-homomorphism, Adams operations [See also 55Q50] 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 55R91: Equivariant fiber spaces and bundles [See also 19L47]

Keywords
$J$–homomorphism Adams operations equivariant $K$–theory equivariant fiber spaces and bundles

Citation

French, Christopher P. The equivariant $J$–homomorphism for finite groups at certain primes. Algebr. Geom. Topol. 9 (2009), no. 4, 1885--1949. doi:10.2140/agt.2009.9.1885. https://projecteuclid.org/euclid.agt/1513797069


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References

  • J F Adams, On the groups $J(X)$. I, Topology 2 (1963) 181–195
    Mathematical Reviews (MathSciNet): MR0159336
    Digital Object Identifier: doi:10.1016/0040-9383(63)90001-6
  • J F Adams, On the groups $J(X)$. II, Topology 3 (1965) 137–171
    Mathematical Reviews (MathSciNet): MR0198468
    Zentralblatt MATH: 0137.16801
    Digital Object Identifier: doi:10.1016/0040-9383(65)90040-6
  • J F Adams, On the groups $J(X)$. III, Topology 3 (1965) 193–222
    Mathematical Reviews (MathSciNet): MR0198469
    Zentralblatt MATH: 0137.16901
    Digital Object Identifier: doi:10.1016/0040-9383(65)90054-6
  • J F Adams, On the groups $J(X)$. IV, Topology 5 (1966) 21–71
    Mathematical Reviews (MathSciNet): MR0198470
    Zentralblatt MATH: 0145.19902
    Digital Object Identifier: doi:10.1016/0040-9383(66)90004-8
  • J F Adams, J-P Haeberly, S Jackowski, J P May, A generalization of the Atiyah–Segal completion theorem, Topology 27 (1988) 1–6
    Mathematical Reviews (MathSciNet): MR935523
    Zentralblatt MATH: 0657.55007
    Digital Object Identifier: doi:10.1016/0040-9383(88)90002-X
  • J Allard, Adams operations in $KO(X)\oplus KSp(X)$, Bol. Soc. Brasil. Mat. 5 (1974) 85–96
    Mathematical Reviews (MathSciNet): MR0372857
    Zentralblatt MATH: 0341.55018
    Digital Object Identifier: doi:10.1007/BF02584775
  • M F Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. $(2)$ 19 (1968) 113–140
    Mathematical Reviews (MathSciNet): MR0228000
    Zentralblatt MATH: 0159.53501
    Digital Object Identifier: doi:10.1093/qmath/19.1.113
  • M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3–38
    Mathematical Reviews (MathSciNet): MR0167985
    Zentralblatt MATH: 0146.19001
    Digital Object Identifier: doi:10.1016/0040-9383(64)90003-5
  • M F Atiyah, D O Tall, Group representations, $\lambda $–rings and the $J$–homomorphism, Topology 8 (1969) 253–297
    Mathematical Reviews (MathSciNet): MR0244387
    Digital Object Identifier: doi:10.1016/0040-9383(69)90015-9
  • R Bott, Lectures on $K(X)$, Math. Lecture Note Ser., W. A. Benjamin, New York-Amsterdam (1969)
    Mathematical Reviews (MathSciNet): MR0258020
  • T tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer, Berlin (1979)
    Mathematical Reviews (MathSciNet): MR551743
    Zentralblatt MATH: 0445.57023
  • A D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275–284
    Mathematical Reviews (MathSciNet): MR690052
    Zentralblatt MATH: 0521.57027
    Digital Object Identifier: doi:10.1090/S0002-9947-1983-0690052-0
  • C French, The equivariant $J$–homomorphism, Homology Homotopy Appl. 5 (2003) 161–212
    Mathematical Reviews (MathSciNet): MR1989617
    Zentralblatt MATH: 1032.55016
    Digital Object Identifier: doi:10.4310/HHA.2003.v5.n1.a8
  • K Hirata, A Kono, On the Bott cannibalistic classes, Publ. Res. Inst. Math. Sci. 18 (1982) 1187–1191
    Mathematical Reviews (MathSciNet): MR688953
    Zentralblatt MATH: 0525.55016
    Digital Object Identifier: doi:10.2977/prims/1195183304
  • J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975) xiii+98
    Mathematical Reviews (MathSciNet): MR0370579
    Zentralblatt MATH: 0321.55033
    Digital Object Identifier: doi:10.1090/memo/0155
  • J P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, Lecture Notes in Math. 577, Springer, Berlin (1977) With contributions by F Quinn, N Ray, and J Tornehave
    Mathematical Reviews (MathSciNet): MR0494077
  • J P May, Equivariant homotopy and cohomology theory, from: “Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981)”, (S Gitler, editor), Contemp. Math. 12, Amer. Math. Soc. (1982) 209–217
    Mathematical Reviews (MathSciNet): MR676330
    Zentralblatt MATH: 0506.55002
  • D Quillen, The Adams conjecture, Topology 10 (1971) 67–80
    Mathematical Reviews (MathSciNet): MR0279804
    Zentralblatt MATH: 0219.55013
    Digital Object Identifier: doi:10.1016/0040-9383(71)90018-8
  • J-P Serre, Linear representations of finite groups, Graduate Texts in Math. 42, Springer, New York (1977) Translated from the second French edition by L L Scott
    Mathematical Reviews (MathSciNet): MR0450380
  • S Waner, Equivariant classifying spaces and fibrations, Trans. Amer. Math. Soc. 258 (1980) 385–405
    Mathematical Reviews (MathSciNet): MR558180
    Zentralblatt MATH: 0447.55011
    Digital Object Identifier: doi:10.1090/S0002-9947-1980-0558180-5

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