Topology formulas for groups and homogeneous spaces

A topology formula needs a named space and a named map. The formulas below concern finite-dimensional Lie groups, specified closed subgroup inclusions, and principal bundles over CW bases. Basepoints are the identity and its coset. The lesson on groups, cosets, and classifying spaces derives the two sphere examples in detail.

Coverings and central quotients

Let $G$ be a connected Lie group and $\Gamma\subset Z(G)$ a discrete central subgroup. The quotient map $q:G\to G/\Gamma$ is a covering. Lifting a based loop in the quotient to a path starting at the identity gives its endpoint in $\Gamma$, and yields

$$1\longrightarrow\pi_1(G) \xrightarrow{q_*}\pi_1(G/\Gamma) \xrightarrow{\mathrm{end}}\Gamma \longrightarrow1.$$

In particular, $\pi_1(G/\Gamma)\cong\Gamma$ when $G$ is simply connected. Without that hypothesis, the kernel remains present. For $n\geq2$,

$$q_*:\pi_n(G)\xrightarrow{\cong}\pi_n(G/\Gamma).$$

The covering is a local diffeomorphism and preserves the Lie algebra. It need not preserve the fundamental group or the permitted group representations. Sources: Conrad, Fundamental groups of Lie groups, pp. 1–2, and Hatcher, Proposition 4.1, p. 342.

Checks. Since $SU(2)\cong S^3$, its quotient by $\{\pm I\}$ is $SO(3)$ with $\pi_1=\mathbb Z_2$ and $\pi_2=0$. In contrast, identify $U(1)/\mu_k$ with $U(1)$ by $[z]\mapsto z^k$, where $\mu_k$ is the subgroup of $k$th roots of unity. The covering becomes $z\mapsto z^k$, giving

$$0\longrightarrow\mathbb Z\xrightarrow{\times k}\mathbb Z \longrightarrow\mathbb Z_k\longrightarrow0.$$

The quotient circle still has fundamental group $\mathbb Z$, not $\mathbb Z_k$. These examples distinguish a covering from a universal covering.

A closed subgroup and its homogeneous space

For a closed subgroup inclusion $i:H\hookrightarrow G$, write $p:G\to G/H$ for the coset projection. It is a locally trivial principal $H$-bundle. The based long exact sequence includes

$$\begin{aligned} \cdots\longrightarrow\pi_2(G)&\xrightarrow{p_*}\pi_2(G/H) \xrightarrow{\partial}\pi_1(H) \xrightarrow{i_*}\pi_1(G)\\ &\xrightarrow{p_*}\pi_1(G/H) \xrightarrow{\partial}\pi_0(H) \longrightarrow\pi_0(G) \longrightarrow\pi_0(G/H)\longrightarrow *. \end{aligned}$$

At the component end, exactness means image equals the preimage of the distinguished point. Do not read that tail as a sequence of abelian groups. If $G$ is connected, $G/H$ is connected and the tail after $\pi_0(H)$ terminates in a point. Sources: Mitchell, §3, p. 5, and Hatcher, Theorem 4.41, p. 376.

The boundary $\partial:\pi_2(G/H)\to\pi_1(H)$ is the clutching loop of the pullback $H$-bundle on the sphere. It depends on the inclusion and on generator conventions. Useful consequences are:

Hypotheses	Consequence
$\pi_2(G)=0$	$\partial$ identifies $\pi_2(G/H)$ with $\ker i_*$
$G,H$ connected	$\pi_1(G/H)\cong\pi_1(G)/\operatorname{im}i_*$
$G$ connected and simply connected, $H$ connected	$\pi_1(G/H)=0$

For disconnected $H$, the second and third rows cannot be used as written. The example $H=\{\pm I\}\subset SU(2)$ has $\pi_1(H)=0$ but $\pi_1(G/H)=\mathbb Z_2$, supplied by $\pi_0(H)$.

Normalized sphere examples

Use $e^{i\theta}\mapsto\operatorname{diag}(e^{i\theta},e^{-i\theta})$ for $U(1)\subset SU(2)$, and the rotation through the actual angle $\varphi$ about the third axis for $SO(2)\subset SO(3)$. Each angle has period $2\pi$ in its own group. Under compatible spin-cover conventions, $\varphi=2\theta$.

Choose the common sphere generator so the Hopf boundary is $+1$. Then

$$\begin{array}{rcl} SU(2)/U(1):&0\longrightarrow\mathbb Z \xrightarrow{\times1}\mathbb Z\longrightarrow0,\\[4pt] SO(3)/SO(2):&0\longrightarrow\mathbb Z \xrightarrow{\times2}\mathbb Z \xrightarrow{\bmod2}\mathbb Z_2\longrightarrow0. \end{array}$$

In each row the first $\mathbb Z$ is $\pi_2(G/H)$ and the second is $\pi_1(H)$. The last nonzero term in the second row is $\pi_1(SO(3))$. Naturality gives the factor $2$: the cover induces the identity between the sphere orbits and winding multiplication by $2$ between the fibers. The Hopf boundary is an isomorphism because $\pi_1(S^3)=\pi_2(S^3)=0$; see Hatcher, p. 375 and Example 4.45.

The SU(2) circle map and SO(3) circle map fix the embeddings. The topology workbench computes these two curated cases; it does not accept an arbitrary subgroup embedding.

Classifying-space shift

For the universal principal bundle $G\to EG\to BG$ with contractible $EG$,

$$\pi_n(BG)\cong\pi_{n-1}(G)\quad(n\geq2), \qquad \pi_1(BG)\cong\pi_0(G).$$

For a CW complex $X$, principal $G$-bundles up to isomorphism over $X$ are classified by unbased homotopy classes $[X,BG]$. For connected $G$, the simply connected space $BG$ therefore gives

$$\{\text{principal }G\text{-bundles on }S^2\}/\cong \ \longleftrightarrow\ \pi_1(G).$$

Sources: Mitchell, Theorem 7.4, p. 15, and Corollary 11.2, p. 23. Thus there is one $SU(2)$ bundle class on $S^2$, two $SO(3)$ bundle classes, and an integer family of $U(1)$ bundle classes. This classification leaves connections, curvature, action, and dynamics unspecified.

For a symmetry-breaking application, first identify the vacuum orbit and the gauge bundle. An $H$ reduction on $S^2$ extends to a trivial $G$-bundle only if its clutching class lies in $\ker i_*$. This condition does not establish the existence or stability of a physical defect. See topological defects and the worked global-form exercises.