On the \(S^ 1\)-Segal conjecture
For a compact Lie group G the Segal conjecture can be formulated similarly to that for finite groups as follows. Let \(\pi^ G_ k(S^ 0)\) be the equivariant stable homotopy group. Let EG be a free contractible G-CW complex and let \(EG^{(r)}\) be the equivariant skeleton. The projection \(EG^{(r)}\to *\) induces a homomorphism \(\pi^ G_ k(S^ 0)\cong \pi_ G^{-k}(S^ 0)\to \pi_ G^{- k}(EG_+^{(r)})\). It is well known that \(\pi_ G^{- k}(EG_+^{(r)})\cong \pi^{-k}((EG^{(r)}/G)_+)\) and we have a homomorphism \(\alpha_ r: \pi_ k^ G(S^ 0)\to \pi^{- k}((EG^{(r)}/G)_+)\). Since \(\lim_{\to}EG^{(r)}/G=BG\), we write \(\lim_{\leftarrow}\pi^{-k}((EG^{(r)}/G)_+)\) as \({\mathcal H}^{- k}(BG;{\mathbb{S}})\), then we have a homomorphism \[ \alpha: \pi^ G_ k(S^ 0)\to {\mathcal H}^{-k}(BG;{\mathbb{S}}). \] The Segal conjecture for nonfinite groups should be stated as follows. Conjecture. \(\alpha\) : \(\pi\) \({}^ G_ k(S^ 0)\to {\mathcal H}^{-k}(BG;{\mathbb{S}})\) has a dense image for \(k\in {\mathbb{Z}}\). For \(k\geq 0\), \textit{M. Feshbach} [Topology, Proc. spec. Seminar, Vol. 3, Mexico 1981-1982, 195-216 (1982; Zbl 0534.55013)] has shown that the conjecture holds for any compact Lie group. Now the purpose of this paper is to prove the following Theorem. The Segal conjecture holds for \(G=S^ 1\). Moreover \(\alpha\) is an isomorphism if \(k\leq 0.\) Our method is an approximation of \(S^ 1\) by finite cyclic groups. For this we use the \(S^ 1\)-transfer and in Section 2 we explain this in more general situation. In Section 3 we show approximation theorems for stable cohomotopy and stable homotopy of \(BS^ 1\), and in Section 4 the proof of the theorem will be given.
scientific article; zbMATH DE number 3907364