The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U:M\to {\text{SU}}_{N_F}$ he thinks of the meson fields as of global sections in a bundle $B(M,{\text{SU}}_{N_F},G)=P(M,G){\times }_G{\text{SU}}_{N_F}$. For calculations within the skyrmion bundle the author introduces by means of the so-called equivariant cohomology an analogue of the topological charge and the Wess-Zumino term. The final result of this paper is the following Theorem. For the skyrmion bundle with $N_F\le 6$, one has $$H^{*}\left(EG{\times }_G{\text{SU}}_{N_F}\right)\cong H^{*}{\left({\text{SU}}_{N_F}\right)}^G\cong \text{S}\left({\underline{G}}^{*}\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right)\cong H^{*}\left(BG\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right),$$ where $EG(BG,G)$ is the universal...

We study secondary obstructions to representing a line bundle as the pull-back of a line bundle on $S^2$ and we interpret them geometrically.