The equivariant \(J\)-homomorphism for finite groups at certain primes
In order to compute the group \(J(X)\), \textit{J. F. Adams} [Topology 3, 137-171 (1965; Zbl 0137.16801)] introduced two groups \(J'(X)\) and \(J''(X)\) equipped with canonical epimorphisms \(J''(X) \to J(X) \to J'(X)\) such that the composite of these epimorphisms induces an identity \(J'(X)=J''(X)\), which of course depends on the solution to the Adams conjecture. The purpose of this paper is to consider this identity in the \(G\)-equivariant setting where \(G\) is a finite group. The above identity is rephrased in the \(J\)-theory diagram of \textit{J. P. May} [``\(E_\infty\) ring spaces and \(E_\infty\) ring spectra'', Lect. Notes Math. 577 (1977; Zbl 0345.55007)]. The central columns of this diagram form a square, which is called here the Adams-May square: \[ \begin{tikzcd} BO \rar["\psi^k-1"]\dar["\sigma^k" '] & B\mathrm{Spin}\dar["\rho^k"] \\ BO_\otimes \rar["\psi^k/1" '] & B\mathrm{Spin}_\otimes \end{tikzcd} \] where \(\sigma^k\) denotes a map which arises from using a space level solution to the Adams conjecture and \(\rho^k\) is a geometric interpretation of the Adams-Bott cannibalistic class. It is asserted there that after localizing at a prime \(p\), for a suitable choice of \(k\), this square becomes a homotopy pullback, which implies immediately that \(J'(X)=J''(X)\). In a previous paper [\textit{C. French}, Homology Homotopy Appl. 5, No.~1, 161--212 (2003; Zbl 1032.55016)] the author constructed an equivariant version of this Adams-May square, and proved that when \(G\) is a \(p\)-group with \(p\neq 2\), for suitable \(k\), after restricting to \(G\)-connected covers and completing at the prime \(p\), this equivariant Adams-May square becomes a homotopy pullback. In this paper, the author considers the case when \(G\) is a finite group such that none of the prime divisors of \(|G|\) are congruent to 1 modulo \(p\) (a prime), so that \(G\) is not a \(p\)-group. Here one restricts oneself to the case when \(p\) is odd for the sake of brevity. The author constructs a natural splitting of the \(p\)-completion of equivariant \(KO\)-theory: \(KO_G(X)_p^\wedge\cong W_G(X)\oplus WO_G^\perp(X)\) which is compatible with the four maps \(\psi^k-1, \;\psi^k/1, \;\sigma^k\) and \(\rho^k\), so that the Adams-May square splits as a product of two squares. It is also shown that every element in \(WO_G^\perp(X)\) can be written as a linear combination of elements of the form \((\psi^\ell-1)(\xi)\) where \(\ell\) is an odd integer relatively prime to \(p\) and \(|G|\), hence these elements have trivial spherical fibrations by the solution of the equivariant Adams conjecture. Then the main result (Theorem 2.10) states (the case \(p=2\) omitted) that if \(k\) is an odd integer such that \((k, |G|)=1\) and which is a topological generator of the units in \(\mathbb{Z}_p^\wedge\), then the square thus obtained \[ \begin{tikzcd} W_G \rar["\psi^k-1"]\dar["\sigma^k" '] & W_G \dar["\rho^k"] \\ W_{G\otimes} \rar["\psi^k/1" '] & W_{G\otimes} \end{tikzcd} \] \noindent is a homotopy pullback. The proof consists of two parts. In the first part it is shown that \(\rho^k\) is an equivariant weak homotoy equivalence (Theorem 4.1) and in the second part it is shown that \(\sigma^k\) and \(\rho^k\) are homotopic (Corollary 6.9). These facts show that the induced map between the fibres of \(\psi^k-1\) and \(\psi^k/1\) becomes an equivariant weak homotopy equivalence, which yields the desired result. Finally it is remarked that this result yields a splitting of the equivariant analogue of \(SF\) at the prime \(p\), similar to that in the \(p\)-group case.
scientific article; zbMATH DE number 5610805
scientific article