examples of fiber bundles

Examples of Fiber Bundles

\(G \to EG \to {\mathbf{B}}G\)
\({\mathbb{Z}}\to {\mathbb{R}}\to S^1\)
\({\mathbb{Z}}^n \to {\mathbb{R}}^n \to T^n\)
\({\mathbb{Z}}^{\ast n} \to ??? \to \bigvee_n S^1\)
\({\mathbb{Z}}_2 \to S^\infty \to {\mathbb{RP}}^\infty\)
\({\mathbb{Z}}_n \to S^\infty \to L_n^\infty\)
\(S^0 \to S^\infty \to {\mathbb{RP}}^\infty\)
\(S^1 \to S^\infty \to {\mathbb{CP}}^\infty\)
\(S^3 \to S^\infty \to {\operatorname{HP}}^\infty\)
NOT TRUE: \(S^7 \to S^\infty \to {\mathbb{OP}}^\infty\)
\(T^n \to ? \to ({\mathbb{CP}}^\infty)^n\)
Stiefel manifold :
- \(O_n \to V_n({\mathbb{R}}^\infty) \to Gr_n({\mathbb{R}}^\infty)\)
\(GL_n({\mathbb{R}}) \to V_n({\mathbb{R}}^\infty) \to Gr_n({\mathbb{R}}^\infty)\)
\(SO_n \to ? \to ?\)
\(Gr_n({\mathbb{R}}^\infty) \to ? \to Gr_n({\mathbb{R}}^\infty)\)
\(\pi_1(\Sigma_g) \to ? \to \Sigma_g\)
\(S_n \to ??? \to \left\{{U \subset {\mathbb{R}}^\infty,~ |U| = n}\right\}\)