Stable splitting of the complex connective \(K\)-theory of \(BO(n)\) (Q664701)

In the article the authors prove a \(2\)-local stable splitting for the spectrum \(bu \wedge BO(n)\), where \(bu\) denotes the connective (real) \(K\)-theory spectrum and \(BO(n)\) denotes the classifying space for the orthogonal group \(O(n)\). More specifically they obtain that \(bu \wedge BO(n)\) is 2-locally homotopy equivalent to a wedge of shifted copies of \(bu\), \(bu \wedge B\mathbb{Z}/2\) and \(H\mathbb{Z}/2\) respectively, where \(B\mathbb{Z}/2\) stands for the classifying space of the group \(\mathbb{Z}/2\), and \(H\mathbb{Z}/2\) stands for the corresponding Eilenberg-MacLane spectrum representing reduced singular homology with coefficients in \(\mathbb{Z}/2\). Indeed within their stable splitting the authors find precisely one shifted copy of \(bu\) for each monomial in the Pontryagin classes of \(BO(n)\), precisely one shifted copy of \(bu \wedge B\mathbb{Z}/2\) for each monomial in the Pontryagin classes of \(BO(n-1)\), and precisely one shifted copy of the Eilenberg-MacLane spectrum \(H\mathbb{Z}/2\) for each monomial \(w_2^{2m_1}w_4^{2m_2}\cdots w_{2t}^{2m_t}\) in the Stiefel-Withney classes with \(\sum_{i=1}^{t}m_i \geq 0\), \(2t \leq n-1\). Conceptually their line of argument is sort of standard: first they establish an associated direct sum composition for the \(\mathbb{Z}/2\)-cohomology of \(bu \wedge BO(n)\), then they find spectra \(X_i\) whose \(\mathbb{Z}/2\)-cohomology is isomorphic to direct summands of the splitting, and finally they construct a map from \(bu \wedge BO(n)\) to the product of the \(X_i\) inducing an isomorphism in \(\mathbb{Z}/2\)-cohomology. More specifically the authors build on a result by the first author who already presented a useful decomposition of the \(\mathbb{Z}/2\)-cohomology of \(bu \wedge BO(n)\) in [\textit{W. S. Wilson}, J. Lond. Math. Soc., II. Ser. 29, 352--366 (1984; Zbl 0521.55005)]. After a suitable change of the direct sum composition given there the individual summands can be identified with the \(\mathbb{Z}/2\)-cohomology of \(bu\), \(bu \wedge B\mathbb{Z}/2\) or \(H\mathbb{Z}/2\) respectively, and the relevant map inducing the desired isomorphism in \(\mathbb{Z}/2\)-cohomology can be constructed from the respective characteristic classes labelling the summands.