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Examples
Examples of Fibrations
- Path-Loop fibration: \(\Omega X\to {\mathcal{P}}X \to X\) involving the Path space.
Covering spaces
- \({\mathbf{Z}}\to {\mathbf{R}}\to S^1\)
- \({\mathbf{Z}}^n \to {\mathbf{R}}^n \to T^n\)
- \({\mathbf{Z}}^{\ast n} \to ??? \to \bigvee_n S^1\)
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\(C_2 \to S^\infty \to {\mathbf{RP}}^\infty\)
- \(C_2 \to S^n \to {\mathbf{RP}}^n\)
- \(C_n \to S^\infty \to L_n^\infty\)
- \(\pi_1(\Sigma_g) \to \widetilde{\Sigma_g} \to \Sigma_g\)
- \(C_2 \to {\operatorname{Spin}}_n \to {\operatorname{SO}}_n\)
- \({\operatorname{U}}_1 \to \mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }_n \to \mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }_n/{\operatorname{U}}_1 \cong {\operatorname{SO}}_n\)
Hopf
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\(S^0 \to S^\infty \to {\mathbf{RP}}^\infty\)
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\(S^1 \to S^\infty \to {\mathbf{CP}}^\infty\)
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\(S^3 \to S^\infty \to {\operatorname{HP}}^\infty\)
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NOT TRUE: \(S^7 \to S^\infty \to {\mathbb{OP}}^\infty\)
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\(T^n \to ? \to ({\mathbf{CP}}^\infty)^n\)
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\(SO_n \to ? \to ?\)
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\(Gr_n({\mathbf{R}}^\infty) \to ? \to Gr_n({\mathbf{R}}^\infty)\)
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\(S_n \to ??? \to \left\{{U \subset {\mathbf{R}}^\infty,~ |U| = n}\right\}\)
Moduli
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Involving frame bundles or the Stiefel manifold
- Taking the linear span: \(V_k({\mathbf{R}}^n) \to {\operatorname{Gr}}_k({\mathbf{R}}^n)\), generalizes \(S^{n-1}\to {\mathbf{RP}}^{n-1}\) for \(k=1\).
- \(V_k({\mathbf{C}}^n) \to {\operatorname{Gr}}_k({\mathbf{C}}^n)\) generalizing the Hopf bundles for \(n-2,k=1\).
- \({\operatorname{O}}_{n-k}({\mathbf{R}}) \to {\operatorname{O}}_n({\mathbf{R}}) \to V_k({\mathbf{R}}^n)\).
- \(O_n \to V_n({\mathbf{R}}^\infty) \to Gr_n({\mathbf{R}}^\infty)\)
- \(GL_n({\mathbf{R}}) \to V_n({\mathbf{R}}^\infty) \to Gr_n({\mathbf{R}}^\infty)\)
Lie groups
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For any \(K\leq H \leq G\), the projection \(G/K\to G/H\).
- \({\operatorname{SU}}_{n-1} \to {\operatorname{SU}}_n \to S^{2n-1}\)
- \({\operatorname{SU}}_n \to {\operatorname{U}}_n \xrightarrow{\operatorname{det}} {\operatorname{U}}_1\)
- \({\operatorname{U}}_{n-1} \to {\operatorname{U}}_n \to {\operatorname{U}}_{n-1}/{\operatorname{U}}_n\cong S^{2n-1}\)
- \({\operatorname{O}}_n \to {\operatorname{O}}_{n+1} \to {\operatorname{O}}_{n+1}/{\operatorname{O}}_n \cong S^{n}\)
- \({\operatorname{SO}}_n \to {\operatorname{SO}}_{n+1} \to {\operatorname{SO}}_{n+1}/{\operatorname{SO}}_n \cong S^n\)
Classifying spaces
- \(G \to {\mathsf{E} G}\to {\mathbf{B}}G\)
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\({{\mathbf{B}}{\operatorname{SO}}}_n \rightarrow {{\mathbf{B}}{\operatorname{O}}}_n \rightarrow {\mathbf{RP}}^{\infty}\)
- Related to Pontrayagin classes
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\(S^{2 n-1} \to {\mathbf{B}}{\operatorname{U}}_{n-1} \stackrel{p}{\rightarrow} {\mathbf{B}}{\operatorname{U}}_n\)
- Used in analyzing Chern classes.
Examples of principal bundles
