Obstructions to Embeddings of Bundles of Matrix Algebras in a Trivial Bundle

We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle $A_k\to X$ whose fiber is a complex matrix algebra $M_k(\mathbb C)$ in a trivial bundle with fiber $M_{kl}(\mathbb C)$ under the assumption that $(k,l)=1$. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group $\mathrm{PGL}_k(\mathbb C)$ of the bundle $A_k$ to $\mathrm{SL}_k(\mathbb C)$, which coincides with the first Chern class $c_1(\xi_k)$ reduced modulo $k$ under the assumption that $A_k\cong\mathrm{End}(\xi_k)$ for some vector $\mathbb C^k$-bundle $\xi_k\to X$. If the first obstruction vanishes, then $A_k\cong\mathrm{End}(\widetilde\xi_k)$ for some vector bundle $\widetilde\xi_k\to X$ with structure group $\mathrm{SL}_k(\mathbb C)$, and the second obstruction is $c_2(\widetilde\xi_k)\operatorname{mod} k \in H^4(X,\mathbb Z/k\mathbb Z)$. Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.