On the algebraic K-cohomology of lens spaces (Q2266475)

Let \(F_ q\) be a field of order \(q=p^ d\) and \(b_{F_ q}\) be the 0- connected spectrum of algebraic K-theory for \(F_ q\). Let \(\ell\) be an odd prime number (\(\ell \neq p)\) and \(L^ n_ 0(\ell)\) be the 2n- skeleton of the standard 2n\(+1\) dimensional lens space mod \(\ell\). The cohomology groups \(b^*_{F_ q}(L^ n_ 0(\ell))\) were studied by \textit{G. Nishida} [J. Math. Kyoto Univ. 23, 211-217 (1983; Zbl 0519.55002)] in a special case. The purpose of this paper is to determine completely the cohomology groups \(b^*_{F_ q}(L^ n_ 0(\ell))\). Let bu be the 1-connected spectrum which represents topological K-theory. Denote by \(\Lambda\) the ring of \(\ell\)-adic integers and by \(X_{\Lambda}\) the \(\ell\)-adic completion of a spectrum X. Using the cohomology exact sequence of the fibration \((b_{F_ q})_{\Lambda}\to bu_{\Lambda}\to^{1-\psi^ q}bu_{\Lambda}\) (where \(\psi^ q\) is the Adams operation), the authors have \(\tilde b^{2k}_{F_ q}(L^ n_ 0(\ell))\cong Ker \alpha\) and \(\tilde b_{F_ q}^{2k+1}(L^ n_ 0(\ell))\cong Co\ker \alpha\) where \(\alpha =(1-q^{-k}\psi^ q)^*: F^ k\tilde K^ 0(L^ n_ 0(\ell))\to F^ k\tilde K^ 0(L^ n_ 0(\ell)),\) \(F^ k\tilde K^ 0(L^ n_ 0(\ell))=Ker(\tilde K^ 0(L^ n_ 0(\ell))\to \tilde K^ 0(L^ k_ 0(\ell))\) if \(0\leq k<n\), \(=\tilde K^ 0(L^ n_ 0(\ell))\) if \(k<0\) and \(=0\) if \(k\geq n\). Combining these with the results on topological K-groups of \(L^ n_ 0(\ell)\), they get the main results.