---
date: 2022-02-23 18:45
modification date: Friday 1st April 2022 16:14:07
title: "examples of fibrations"
aliases: [examples of fibrations, examples of fiber bundles, examples of G bundles, examples of principal bundles]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #examples #homotopy/bundles 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# Examples

## Examples of Fibrations


- Path-Loop fibration: $\Omega X\to \mcp X \to X$ involving the [Path space](Path%20space).
- 


# Covering spaces

- $\ZZ \to \RR \to S^1$
- $\ZZ^n \to \RR^n \to T^n$
- $\ZZ^{\ast n} \to ??? \to \bigvee_n S^1$
- $C_2 \to S^\infty \to \RP^\infty$
	- $C_2 \to S^n \to \RP^n$
- $C_n \to S^\infty \to L_n^\infty$
- $\pi_1(\Sigma_g) \to \widetilde{\Sigma_g} \to \Sigma_g$
- $C_2 \to \Spin_n \to \SO_n$
- $\U_1 \to \Spinc_n \to \Spinc_n/\U_1 \cong \SO_n$
  
# Hopf

- $S^0 \to S^\infty \to \RP^\infty$
- $S^1 \to S^\infty \to \CP^\infty$
- $S^3 \to S^\infty \to \HP^\infty$
- NOT TRUE: $S^7 \to S^\infty \to \OP^\infty$

- $T^n \to ? \to (\CP^\infty)^n$


- $SO_n \to ? \to ?$
- $Gr_n(\RR^\infty) \to ? \to Gr_n(\RR^\infty)$

- $S_n \to ??? \to \theset{U \subset \RR^\infty,~ |U| = n}$


# Moduli

- Involving frame bundles or the [Stiefel manifold](Stiefel%20manifold.md) 
	- Taking the linear span: $V_k(\RR^n) \to \Gr_k(\RR^n)$, generalizes $S^{n-1}\to \RP^{n-1}$ for $k=1$.
	- $V_k(\CC^n) \to \Gr_k(\CC^n)$ generalizing the Hopf bundles for $n-2,k=1$.
	- $\Orth_{n-k}(\RR) \to \Orth_n(\RR) \to V_k(\RR^n)$.
	- $O_n \to V_n(\RR^\infty) \to Gr_n(\RR^\infty)$
- $GL_n(\RR) \to V_n(\RR^\infty) \to Gr_n(\RR^\infty)$


# Lie groups

- For any $K\leq H \leq G$, the projection $G/K\to G/H$.
![](attachments/Pasted%20image%2020220401161352.png)
- $\SU_{n-1} \to \SU_n \to S^{2n-1}$
- $\SU_n \to \U_n \mapsvia{\det} \U_1$
- $\U_{n-1} \to \U_n \to \U_{n-1}/\U_n\cong S^{2n-1}$
- $\Orth_n \to \Orth_{n+1} \to \Orth_{n+1}/\Orth_n \cong S^{n}$
- $\SO_n \to \SO_{n+1}  \to \SO_{n+1}/\SO_n \cong  S^n$

# Classifying spaces

- $G \to \EG \to \B G$
- $\BSO_n \rightarrow \BO_n \rightarrow \RP^{\infty}$
	- Related to [Pontrayagin classes](Unsorted/Pontrayagin%20class.md)
- $S^{2 n-1} \to \BU_{n-1} \stackrel{p}{\rightarrow} \BU_n$
	- Used in analyzing [Chern classes](Unsorted/Chern%20class.md).

# Examples of principal bundles

![](attachments/Pasted%20image%2020220403174140.png)