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Examples
Examples of Fibrations
- For any \(K\leq H \leq G\), the projection \(G/K\to G/H\).
- \(G \to EG \to {\mathbf{B}}G\)
- \({\mathbb{Z}}\to {\mathbb{R}}\to S^1\)
- \({\mathbb{Z}}^n \to {\mathbb{R}}^n \to T^n\)
- \({\mathbb{Z}}^{\ast n} \to ??? \to \bigvee_n S^1\)
- \({\mathbb{Z}}_2 \to S^\infty \to {\mathbb{RP}}^\infty\)
- \({\mathbb{Z}}_n \to S^\infty \to L_n^\infty\)
- \(S^0 \to S^\infty \to {\mathbb{RP}}^\infty\)
- \(S^1 \to S^\infty \to {\mathbb{CP}}^\infty\)
- \(S^3 \to S^\infty \to {\operatorname{HP}}^\infty\)
- NOT TRUE: \(S^7 \to S^\infty \to {\mathbb{OP}}^\infty\)
- \(T^n \to ? \to ({\mathbb{CP}}^\infty)^n\)
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Involving frame bundles or the Stiefel manifold
- Taking the linear span: \(V_k({\mathbb{R}}^n) \to {\operatorname{Gr}}_k({\mathbb{R}}^n)\), generalizes \(S^{n-1}\to {\mathbb{RP}}^{n-1}\) for \(k=1\).
- \(V_k({\mathbb{C}}^n) \to {\operatorname{Gr}}_k({\mathbb{C}}^n)\) generalizing the Hopf bundles for \(n-2,k=1\).
- \({\operatorname{O}}_{n-k}({\mathbb{R}}) \to {\operatorname{O}}_n({\mathbb{R}}) \to V_k({\mathbb{R}}^n)\).
- \(O_n \to V_n({\mathbb{R}}^\infty) \to Gr_n({\mathbb{R}}^\infty)\)
- \(GL_n({\mathbb{R}}) \to V_n({\mathbb{R}}^\infty) \to Gr_n({\mathbb{R}}^\infty)\)
- \(SO_n \to ? \to ?\)
- \(Gr_n({\mathbb{R}}^\infty) \to ? \to Gr_n({\mathbb{R}}^\infty)\)
- \(\pi_1(\Sigma_g) \to ? \to \Sigma_g\)
- \(S_n \to ??? \to \left\{{U \subset {\mathbb{R}}^\infty,~ |U| = n}\right\}\)
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\(S^{2 n-1} \to {\operatorname{BU}}_{n-1} \stackrel{p}{\rightarrow} {\operatorname{BU}}_n\)
- Used in analyzing Chern classes.
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\({{\mathbf{B}}{\operatorname{SO}}}_n \rightarrow {{\mathbf{B}}{\operatorname{O}}}_n \rightarrow {\mathbb{RP}}^{\infty}\)
- Related to Pontrayagin classes
Examples of principal bundles
