Obstructions to embeddings of bundles of matrix algebras in a trivial bundle (Q2436383)

Let us consider a locally trivial bundle \(A_k \overset{p_k}\longrightarrow X\) over a compact space \(X\), with fibre the complex matrix algebra \(M_k(\mathbb{C})\). The paper concerns the existence of fibre-preserving unital embeddings \(\mu : A_k \longrightarrow X \times M_n(\mathbb{C})\) over \(X\), where the fibre of the trivial bundle \(X \times M_n(\mathbb{C})\) over \(X\) is \(M_{kl}(\mathbb{C})\) and \((k,l) = 1\). To achieve this goal, the author calculates cohomology obstructions in the following way: first he reduces the embedding problem for bundles of matrix algebras into a trivial bundle to the problem of constructing a section in some bundle. This is interesting because in small dimensions these obstructions can be given in terms of well-known characteristic classes, which are easy to calculate. The first obstruction turns out to be the obstruction to the reduction of the structure group of \(PGL_k(\mathbb{C})\) of the considered bundle to \(SL_k(\mathbb{C})\). If the first obstruction vanishes, then the second obstruction is defined, which is a Chern class of some \(\mathbb{C}^k\)-vector bundle. Finally, a general construction is given of the obstructions by using the Postnikov tower.