The Morava \(K\)-theory of spaces related to \(BO\)
Let \({bo}_*\) denote the \(\Omega\)-spectrum with \({bo}_0 = {\mathbb Z} \times BO\) and \({BG}_*\) denote the \(\Omega\)-spectrum with \({BG}_0 = BG\) for \(G = O\), \(SO\) or \(Spin\). This paper computes the Morava \(K\)-theory at the prime \(2\), denoted \(K_*(\enskip)\), of the spaces in these \(\Omega\) spectra. These spaces are related by a sequence of connective covering maps and so these computations can be done inductively by studying such coverings. The authors show that if \(B = {bo}_i\) with \(i \geq 4\) or \(B= {BG}_i\) with \(i \geq 2\) and \(E \to B\) is a connective cover with fibre \(F\) then there is a short exact sequence of Hopf algebras \(K_* \to K_* F \to K_* E \to K_* B \to K_*\). The authors also describe what happens when \(B = {bo}_i\) with \(i < 4\) or \(B = BG_i\) with \(i < 2\) (the situation here is sufficiently varied and complicated that the statement of results takes up more than 3 pages of the paper). Of particular interest is their description of the Morava \(K\)-theory of the map \({bo}_8 \to {BSpin}_0\) which fits into a 6-term exact sequence of Hopf algebras. Results corresponding to these for \(BU\) were previously established by \textit{D. C. Ravenel, W. S. Wilson} and \textit{N. Yagita} [K-theory 15, 149-199 (1998; Zbl 0912.55002)]. In the present paper the authors identify the Hopf algebra cokernel of the forgetful map \(BU \to BO\) and show that it is isomorphic to \(K_* (\Pi_{i >0} K({\mathbb F}_2,i))\). They also use their results to compute the 2-adicly completed Brown-Peterson cohomology of many of the spaces in these \(\Omega\)-spectra.
scientific article; zbMATH DE number 2119097
scientific article