On images of the fixed-point homomorphism in the Z/p-equivariant stable homotopy groups
Let \(G={\mathbb{Z}}/p\) be a cyclic group of odd prime order, and let V be a faithful one-dimensional complex representation. The author identifies the image of the fixed point map \(\phi: \pi\) \({}^ G_{nV}(S^ 0)\to \pi_ 0(S^ 0)\cong {\mathbb{Z}}\) of stable homotopy groups. When \(n\leq 0\), \(\phi\) is obviously surjective, and when \(n\geq 1\), the image is \(p^{1+[n/p-1]}{\mathbb{Z}}\). This is the analogue for odd primes of a result of \textit{D. S. Landweber} for \(p=2\) [Ann. Math., II. Ser. 89, 125-137 (1969; Zbl 0174.262)] and the method is similar. To show the image lies in \(p^{1+[n/p-1]}{\mathbb{Z}}\) the author uses equivariant K-theory: if \(n\not\equiv 0 mod p-1\), it is immediate from the formula for the Euler class of nV, and if \(n\equiv 0 mod p-1\), this is augmented by use of the Adams operation \(\psi^ p\). To see that \(p^{1+[n/p-1]}\in im(\phi)\) the author notes that \(\phi\) lies in an exact sequence whose next term is \(\pi^ G_{nV}(\Sigma S(nV)_+)\). By finiteness of \(K_ G(S(nV)_+)\) this is isomorphic to a homotopy group in integer degrees, and hence to a nonequivariant homotopy group of a stunted lens space. Now the author can use his earlier result on the order of the identity map of such spaces [Jap. J. Math., New Ser. 10, 1-8 (1984; Zbl 0602.55008)]. \(\{\) In Lemma 6, s and r should be specified by \(m-1=s(p-1)+r\) (0\(\leq r\leq p-2):\) the exceptional case needing \(\psi^ p\) is then \(r=p-2\}\).
faithful one-dimensional complex representation
scientific article; zbMATH DE number 4159629