---
date: 2022-04-13 09:06
modification date: Wednesday 13th April 2022 09:06:56
title: "examples of classifying spaces"
aliases: [examples of classifying spaces]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- <https://mathoverflow.net/questions/56363/list-of-classifying-spaces-and-covers>
- Links:
	- #todo/create-links 

---

# examples of classifying spaces



- $\B B_n = \Conf_n(\RR^2)$ for $B_n$ the [pure braid group](Braid%20group)
	- Take the *unordered* [configuration space](Unsorted/configuration%20space.md) for the usual braid group.
- $\B \Free(S) \cong \bigvee_{S} S^1$.
- $\GL_n(\RR) \to V_n \to \B \GL_n(\RR) \cong \Gr_n(\RR^\infty)$ for $V_n$ the [Stiefel manifold](Unsorted/Stiefel%20manifold.md).
	- Note $\B\Orth_n(\RR) = \B\GL_n(\RR)$.
	- Classifies rank $n$ vector bundles.
- $\B S^3 = \B \Sp_1 = \B\SU_2 = B \SS(H) = \HP^\infty$ where $H$ are [quaternions](quaternions) and $\SS(H)$ denotes the units.
- $\B \pi_1 \Sigma_g \cong \Sigma_g$ for $g\geq 1$ (so [hyperbolic surfaces](hyperbolic%20surfaces)).
	- #todo/why
- $\B \pi_1 M \cong M$ for $(M, g)$ Riemannian if the sectional curvature satisfies $\sec _g M \leq  0$.
	- #todo/why
- $\B(G\ast H) \cong \B G \wedgeprod \B H$
- $\B \Aut_\Top(M)$ classifies fiber bundles with fiber $M$
- $\B S_n$ classifies $n\dash$sheeted covering spaces

## Building blocks for finite simple groups

- $\ZZ \to \RR \to S^1 \implies \B\ZZ = S^1 = \U_1 = K(\ZZ, 1)$.
	- $\B S^1 =\B \U_1 =  \B^2 \ZZ = \B\GL_1(\RR^\infty) K(\ZZ, 2) = \CP^\infty$.
	- Note $\SL_2(\RR), \SO_2(\RR),, \SO_2(\CC), \Sp_2(\RR), \SU_{1, 1}, \SU_1$ are all homotopy equivalent to $\U_1$.
- $\ZZ\cartpower{n} \to \RR^n\to T^n \implies \B \ZZ\cartpower n \cong (S^1)\cartpower{n}$.
- $C_n \to S^\infty \mapsvia{\vector z \mapsto \zeta_n \vector z} L_n^\infty \implies \B C_n \cong L_n^\infty$, an infinite [lens space](lens%20space).
	- $C_2 \to S^\infty \to \RP^\infty\implies \B C_2 \cong \RP^\infty$
	- $\B \Orth_1(\RR) =\B \Spin_1 = \B \GL_1(\RR) =  K(C_2, 1)$