On complex structures in 8-dimensional vector bundles
Let \(\xi\) be an oriented real \(8\)-plane bundle over a closed connected oriented smooth \(8\)-dimensional manifold \(M\). Supposing that the Stiefel-Whitney classes \(w_2(\xi)\) and \(w_2(M)\) vanish, the authors reduce the question of whether \(\xi\) has a complex structure to the problem of whether a certain associated \(8\)-plane bundle contains an oriented \(2\)-plane bundle. Then they use results from their earlier paper [Colloq. Math. 70, No. 1, 25-40 (1996; Zbl 0844.57026)] to derive explicit necessary and sufficient conditions for the existence of complex structures on \(\xi\) in terms of the cohomology ring of \(M\) and characteristic classes of \(\xi\) and \(M\). Without requiring that \(w_2(M) = 0\), \textit{T. Heaps} [Topology 9, 111-119 (1970; Zbl 0187.20405)] found similar necessary and sufficient conditions for the existence of complex structures on the tangent bundle of \(M\).
scientific article; zbMATH DE number 1143110