## Algebraic & Geometric Topology

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

Christopher P French

#### 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
Revised: 17 July 2009
Accepted: 3 August 2009
First available in Project Euclid: 20 December 2017

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

#### 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

#### References

• J F Adams, On the groups $J(X)$. I, Topology 2 (1963) 181–195
• J F Adams, On the groups $J(X)$. II, Topology 3 (1965) 137–171
• J F Adams, On the groups $J(X)$. III, Topology 3 (1965) 193–222
• J F Adams, On the groups $J(X)$. IV, Topology 5 (1966) 21–71
• J F Adams, J-P Haeberly, S Jackowski, J P May, A generalization of the Atiyah–Segal completion theorem, Topology 27 (1988) 1–6
• J Allard, Adams operations in $KO(X)\oplus KSp(X)$, Bol. Soc. Brasil. Mat. 5 (1974) 85–96
• M F Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. $(2)$ 19 (1968) 113–140
• M F Atiyah, R Bott, A Shapiro, Clifford modules, Topology 3 (1964) 3–38
• M F Atiyah, D O Tall, Group representations, $\lambda$–rings and the $J$–homomorphism, Topology 8 (1969) 253–297
• R Bott, Lectures on $K(X)$, Math. Lecture Note Ser., W. A. Benjamin, New York-Amsterdam (1969)
• T tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer, Berlin (1979)
• A D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275–284
• C French, The equivariant $J$–homomorphism, Homology Homotopy Appl. 5 (2003) 161–212
• K Hirata, A Kono, On the Bott cannibalistic classes, Publ. Res. Inst. Math. Sci. 18 (1982) 1187–1191
• J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975) xiii+98
• 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
• 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
• D Quillen, The Adams conjecture, Topology 10 (1971) 67–80
• 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
• S Waner, Equivariant classifying spaces and fibrations, Trans. Amer. Math. Soc. 258 (1980) 385–405