examples of classifying spaces

\({\mathbf{B}}B_n = {\mathrm{Conf}}_n({\mathbf{R}}^2)\) for \(B_n\) the pure braid group
- Take the unordered configuration space for the usual braid group.
\({\mathbf{B}}\mathsf{Free}(S) \cong \bigvee_{S} S^1\).
\(\operatorname{GL}_n({\mathbf{R}}) \to V_n \to {\mathbf{B}}\operatorname{GL}_n({\mathbf{R}}) \cong {\operatorname{Gr}}_n({\mathbf{R}}^\infty)\) for \(V_n\) the Stiefel manifold.
- Note \({\mathbf{B}}{\operatorname{O}}_n({\mathbf{R}}) = {\mathbf{B}}\operatorname{GL}_n({\mathbf{R}})\).
- Classifies rank \(n\) vector bundles.
\({\mathbf{B}}S^3 = {\mathbf{B}}{\mathsf{Sp}}_1 = {\mathbf{B}}{\operatorname{SU}}_2 = B {\mathbb{S}}(H) = {\operatorname{HP}}^\infty\) where \(H\) are quaternions and \({\mathbb{S}}(H)\) denotes the units.
\({\mathbf{B}}\pi_1 \Sigma_g \cong \Sigma_g\) for \(g\geq 1\) (so hyperbolic surfaces).
- #todo/why
\({\mathbf{B}}\pi_1 M \cong M\) for \((M, g)\) Riemannian if the sectional curvature satisfies \(\sec _g M \leq 0\).
- #todo/why
\({\mathbf{B}}(G\ast H) \cong {\mathbf{B}}G \vee{\mathbf{B}}H\)
\({\mathbf{B}}\mathop{\mathrm{Aut}}_{\mathsf{Top}}(M)\) classifies fiber bundles with fiber \(M\)
\({\mathbf{B}}S_n\) classifies \(n{\hbox{-}}\)sheeted covering spaces

Building blocks for finite simple groups

\({\mathbf{Z}}\to {\mathbf{R}}\to S^1 \implies {\mathbf{B}}{\mathbf{Z}}= S^1 = {\operatorname{U}}_1 = K({\mathbf{Z}}, 1)\).
- \({\mathbf{B}}S^1 ={\mathbf{B}}{\operatorname{U}}_1 = {\mathbf{B}}^2 {\mathbf{Z}}= {\mathbf{B}}\operatorname{GL}_1({\mathbf{R}}^\infty) K({\mathbf{Z}}, 2) = {\mathbf{CP}}^\infty\).
- Note \({\operatorname{SL}}_2({\mathbf{R}}), {\operatorname{SO}}_2({\mathbf{R}}),, {\operatorname{SO}}_2({\mathbf{C}}), {\mathsf{Sp}}_2({\mathbf{R}}), {\operatorname{SU}}_{1, 1}, {\operatorname{SU}}_1\) are all homotopy equivalent to \({\operatorname{U}}_1\).
\({\mathbf{Z}}{ {}^{ \scriptscriptstyle\times^{n} } } \to {\mathbf{R}}^n\to T^n \implies {\mathbf{B}}{\mathbf{Z}}{ {}^{ \scriptscriptstyle\times^{n} } } \cong (S^1){ {}^{ \scriptscriptstyle\times^{n} } }\).
\(C_n \to S^\infty \xrightarrow{\mathbf{z} \mapsto \zeta_n \mathbf{z}} L_n^\infty \implies {\mathbf{B}}C_n \cong L_n^\infty\), an infinite lens space.
- \(C_2 \to S^\infty \to {\mathbf{RP}}^\infty\implies {\mathbf{B}}C_2 \cong {\mathbf{RP}}^\infty\)
- \({\mathbf{B}}{\operatorname{O}}_1({\mathbf{R}}) ={\mathbf{B}}{\operatorname{Spin}}_1 = {\mathbf{B}}\operatorname{GL}_1({\mathbf{R}}) = K(C_2, 1)\)