Covering Spaces and the Fundamental Group of the Rotation Group
This appendix proves Corollary 18.38 of Lie Groups, Lie Algebras, and Fibre Bundles: that \(\pi_{1}\left(\SO(3,\R)\right) \cong \Z_{2}\), and more generally that when a simply connected topological group covers another one, the fundamental group of the base is the kernel of the covering homomorphism. Theorem 18.37 already supplies everything algebraic — that \(\Phi : \SU(2) \rightarrow \SO(3,\R)\) is a surjective homomorphism with kernel \(\set{\identity,-\identity}\) — and Proposition 18.33 supplies the topology of the total space, that \(\SU(2) \cong S^{3}\) is compact, connected and simply connected. What is missing, and is built here, is the bridge between the two: the covering-space machinery that converts a discrete kernel upstairs into a fundamental group downstairs.
Only the material of Topological and Metric Spaces is assumed: open sets, continuity, connectedness, compactness (Theorem 10.11), and simple connectedness (Definition 10.17). That chapter already carries the technique in its simplest instance. The continuous argument along a path of Lemma 10.19 is precisely the unique lifting of a path through the covering \(\R \rightarrow \C\setminus\set{0}\), \(\theta \mapsto \ee^{\ii\theta}\) composed with a radius, and the local-constancy argument of Proposition 10.23 is the homotopy invariance of the lift in that instance. The two lemmas below are those two arguments written for an arbitrary covering; the reader who has followed Section 10.1.5 has seen the whole idea already, and what is added here is generality, not a new device.
Statement
A continuous surjection \(p : E \longrightarrow B\) of topological spaces is a covering map if every \(b \in B\) has an open neighbourhood \(V\) that is evenly covered: \(p^{-1}(V)\) is a union of pairwise disjoint open sets \(\set{W_{\alpha}}\) — the sheets over \(V\) — each of which \(p\) maps homeomorphically onto \(V\). The set \(p^{-1}(b)\) is the fibre over \(b\); a lift of a continuous map \(g : X \longrightarrow B\) is a continuous \(\tilde{g} : X \longrightarrow E\) with \(p \circ \tilde{g} = g\). Rests on Definitions 10.2 and 10.6.
Let \(X\) be a topological space and \(x_{0} \in X\). A loop at \(x_{0}\) is a continuous \(\gamma : [0,1] \longrightarrow X\) with \(\gamma(0) = \gamma(1) = x_{0}\). Two loops \(\gamma_{0}, \gamma_{1}\) at \(x_{0}\) are path homotopic, written \(\gamma_{0} \simeq \gamma_{1}\), if there is a continuous \(H : [0,1]\times[0,1] \longrightarrow X\) with
The concatenation \(\gamma \cdot \delta\) is the loop equal to \(\gamma(2s)\) for \(s \le 1/2\) and to \(\delta(2s-1)\) for \(s \ge 1/2\), and the reverse is \(\bar{\gamma}(s) = \gamma(1-s)\). The set of path-homotopy classes of loops at \(x_{0}\) with the product \([\gamma][\delta] = [\gamma\cdot\delta]\) is the fundamental group \(\pi_{1}(X,x_{0})\). A path-connected \(X\) is simply connected in the sense of Definition 10.17 exactly when \(\pi_{1}(X,x_{0})\) is trivial. Rests on Definitions 10.15 and 10.17.
Let \(p : E \longrightarrow B\) be a covering map with \(E\) path-connected and simply connected, \(B\) path-connected, and fix \(e_{0} \in E\), \(b_{0} = p(e_{0})\). Then the monodromy map
where \(\tilde{\gamma}\) is the unique lift of \(\gamma\) with \(\tilde{\gamma}(0) = e_{0}\), is a well-defined bijection. Rests on Definitions A33.1 and A33.2.
Let \(E\) and \(B\) be topological groups, \(E\) path-connected and simply connected, and let \(p : E \longrightarrow B\) be a surjective continuous homomorphism that is also a covering map, with kernel \(N = p^{-1}(1_{B})\). Then \(N\) is a discrete central subgroup of \(E\), the monodromy map Equation (A33.2) based at \(e_{0} = 1_{E}\) is a group isomorphism
and the group of deck transformations of \(p\) — the homeomorphisms \(\varphi\) of \(E\) with \(p\circ\varphi = p\) — consists exactly of the left translations by elements of \(N\), so that it too is isomorphic to \(N\) and hence to \(\pi_{1}(B,1_{B})\). Rests on Theorem A33.3 and Definition A33.1.
The proof occupies the rest of this section: two lifting lemmas (Two lifting lemmas), the monodromy theorem (The monodromy theorem), the group case (Covering homomorphisms and deck transformations), and the application to \(\SU(2) \rightarrow \SO(3,\R)\) (The rotation group).
Two lifting lemmas
Every lifting argument in covering-space theory is the same argument: chop the domain into pieces small enough to sit inside an evenly covered neighbourhood, lift piece by piece using the sheet that contains the value already fixed, and check that the pieces agree. The chopping is supplied by the following standard consequence of compactness, which Lemma 10.19 used in the disguised form of a partition of mesh smaller than a modulus of uniform continuity.
Let \(K \subset \R^{N}\) be closed and bounded and let \(\set{U_{i}}\) be an open cover of \(K\). There is \(\delta > 0\) such that every subset of \(K\) of diameter smaller than \(\delta\) is contained in a single \(U_{i}\). Rests on Theorem 10.11 and Definition 10.2.
Derives Lemma A33.5. For each \(x \in K\) choose \(i(x)\) with \(x \in U_{i(x)}\) and then \(r(x) > 0\) with \(B_{2r(x)}(x) \subseteq U_{i(x)}\), which is possible because \(U_{i(x)}\) is open. The balls \(B_{r(x)}(x)\) cover \(K\), which is compact by Theorem 10.11, so finitely many of them suffice, say those centred at \(x_{1},\ldots,x_{k}\); put \(\delta = \min_{j} r(x_{j}) > 0\). Let \(S \subseteq K\) have diameter smaller than \(\delta\) and pick \(y \in S\). Then \(y \in B_{r(x_{j})}(x_{j})\) for some \(j\), and every \(z \in S\) satisfies
so \(S \subseteq B_{2r(x_{j})}(x_{j}) \subseteq U_{i(x_{j})}\).
∎Let \(p : E \longrightarrow B\) be a covering map, let \(\gamma : [0,1] \longrightarrow B\) be continuous, and let \(e \in E\) satisfy \(p(e) = \gamma(0)\). There is exactly one continuous \(\tilde{\gamma} : [0,1] \longrightarrow E\) with \(p\circ\tilde{\gamma} = \gamma\) and \(\tilde{\gamma}(0) = e\). Rests on Definition A33.1 and Lemma A33.5.
Derives Lemma A33.6. Existence. Cover \(B\) by evenly covered open sets \(V\) and pull them back: the sets \(\gamma^{-1}(V)\) form an open cover of \([0,1]\), which is closed and bounded, so Lemma A33.5 supplies a \(\delta > 0\) and with it a partition \(0 = t_{0} < t_{1} < \cdots < t_{M} = 1\) of mesh smaller than \(\delta\) such that each \(\gamma\left([t_{j-1},t_{j}]\right)\) lies in a single evenly covered \(V_{j}\).
Define \(\tilde{\gamma}\) inductively. Set \(\tilde{\gamma}(t_{0}) = e\). Suppose \(\tilde{\gamma}\) has been defined and is continuous on \([0,t_{j-1}]\) with \(p\circ\tilde{\gamma} = \gamma\) there. The point \(\tilde{\gamma}(t_{j-1})\) lies in \(p^{-1}(V_{j})\), hence in exactly one sheet \(W\) over \(V_{j}\), and \(p|_{W} : W \longrightarrow V_{j}\) is a homeomorphism. Put
This is continuous on \([t_{j-1},t_{j}]\), satisfies \(p\circ\tilde{\gamma} = \gamma\) there, and agrees at \(t_{j-1}\) with the value already assigned, because \(\tilde{\gamma}(t_{j-1}) \in W\) and \(p|_{W}\) is injective. Two continuous functions on the closed intervals \([0,t_{j-1}]\) and \([t_{j-1},t_{j}]\) agreeing at the common endpoint patch to a continuous function on the union, so the induction proceeds and terminates at \(t_{M} = 1\).
Uniqueness. Let \(\tilde{\gamma}\) and \(\tilde{\gamma}'\) be two lifts with the same initial value, and let \(A = \set{t \in [0,1] \mid \tilde{\gamma}(t) = \tilde{\gamma}'(t)}\), which contains \(0\). Fix \(t \in [0,1]\), let \(V\) be an evenly covered neighbourhood of \(\gamma(t)\), and let \(W, W'\) be the sheets over \(V\) containing \(\tilde{\gamma}(t)\) and \(\tilde{\gamma}'(t)\). The set \(O = \tilde{\gamma}^{-1}(W) \cap (\tilde{\gamma}')^{-1}(W')\) is an open neighbourhood of \(t\) in \([0,1]\). If \(t \in A\) then \(W = W'\), the sheets being disjoint, and on \(O\) both lifts equal \(\left(p|_{W}\right)^{-1}\circ\gamma\), so \(O \subseteq A\): the set \(A\) is open. If \(t \notin A\) then \(W \neq W'\), since a common sheet would force \(\tilde{\gamma}(t) = \tilde{\gamma}'(t)\) by injectivity of \(p|_{W}\); the sheets being disjoint, \(O\) misses \(A\) entirely, so the complement of \(A\) is open too. Thus \(A\) is a nonempty subset of \([0,1]\) that is both open and closed, and \([0,1]\) is connected (Theorem 10.14), so \(A = [0,1]\).
∎Let \(p : E \longrightarrow B\) be a covering map, let \(H : [0,1]\times[0,1] \longrightarrow B\) be continuous, and let \(e \in E\) satisfy \(p(e) = H(0,0)\). There is exactly one continuous \(\tilde{H} : [0,1]\times[0,1] \longrightarrow E\) with \(p\circ\tilde{H} = H\) and \(\tilde{H}(0,0) = e\). If moreover \(H\) is a path homotopy in the sense of Equation (A33.1) — so that \(H(0,t)\) and \(H(1,t)\) are constant in \(t\) — then \(\tilde{H}(0,t)\) and \(\tilde{H}(1,t)\) are constant in \(t\) as well. Rests on Lemmas A33.5 and A33.6.
Derives Lemma A33.7. Existence. As before, the sets \(H^{-1}(V)\) with \(V\) evenly covered form an open cover of the square \([0,1]^{2}\), which is closed and bounded in \(\R^{2}\); Lemma A33.5 supplies \(\delta > 0\), and a grid \(0 = s_{0} < \cdots < s_{M} = 1\), \(0 = t_{0} < \cdots < t_{M} = 1\) of mesh smaller than \(\delta/\sqrt{2}\) makes each closed cell \(R_{jk} = [s_{j-1},s_{j}]\times[t_{k-1},t_{k}]\) have diameter smaller than \(\delta\), hence \(H(R_{jk})\) contained in a single evenly covered \(V_{jk}\).
Order the cells lexicographically, bottom row left to right, then the next row, and lift them one at a time. At each stage the part of the boundary of the current cell \(R\) on which \(\tilde{H}\) has already been defined — call it \(C\) — is the union of at most the left edge and the bottom edge of \(R\), together with the corner joining them, and is therefore connected. (For the very first cell \(C\) is the single point \((s_{0},t_{0})\), where \(\tilde{H} = e\) is prescribed.) The set \(\tilde{H}(C)\) is connected and lies in \(p^{-1}(V)\) for the evenly covered \(V \supseteq H(R)\), which is the disjoint union of open sheets; a connected subset of a disjoint union of open sets lies in one of them, so \(\tilde{H}(C) \subseteq W\) for a single sheet \(W\). Define \(\tilde{H} = \left(p|_{W}\right)^{-1}\circ H\) on \(R\). This is continuous on \(R\), projects to \(H\), and agrees on \(C\) with what was there — again because \(p|_{W}\) is injective and both values lie in \(W\). The extended function is continuous, being continuous on each of finitely many closed cells and consistent on the overlaps. After the last cell \(\tilde{H}\) is defined on the whole square.
Uniqueness. Verbatim the argument of Lemma A33.6: the agreement set of two lifts is open and closed, and the square is connected, being path-connected (Proposition 10.16).
The vertical edges. Suppose \(H(0,t) = b_{0}\) for every \(t\). Then \(t \mapsto \tilde{H}(0,t)\) is a lift of the constant path at \(b_{0}\); so is the constant map \(t\mapsto \tilde{H}(0,0)\); the two agree at \(t = 0\), so by the uniqueness clause of Lemma A33.6 they agree throughout. The same applies at \(s = 1\).
∎The monodromy theorem
The elementary bookkeeping behind Definition A33.2 is recorded first, because it is what makes \(\pi_{1}\) a group at all and because the injectivity of \(\Psi\) uses it.
Let \(\alpha,\beta,\eta\) be paths in a space \(X\) with \(\alpha(1) = \beta(0)\) and \(\beta(1) = \eta(0)\), let \(c_{x}\) denote the constant path at \(x\), and write \(\simeq\) for homotopy with endpoints held fixed. Then
Consequently \(\pi_{1}(X,x_{0})\) of Definition A33.2 is a group, with \([\gamma]^{-1} = [\bar{\gamma}]\); and a continuous \(p : E \longrightarrow B\) with \(p(e_{0}) = b_{0}\) induces a group homomorphism \(p_{*} : \pi_{1}(E,e_{0}) \longrightarrow \pi_{1}(B,b_{0})\), \(p_{*}[\sigma] = [p\circ\sigma]\). Rests on Definition A33.2.
Derives Lemma A33.8. All three relations of Equation (A33.5) are reparametrizations. If \(\phi : [0,1] \rightarrow [0,1]\) is continuous with \(\phi(0) = 0\) and \(\phi(1) = 1\), then for any path \(\mu\) the map \(H(s,t) = \mu\left((1-t)\phi(s) + ts\right)\) is continuous, fixes the endpoints, and joins \(\mu\circ\phi\) to \(\mu\); so \(\mu\circ\phi \simeq \mu\). The two sides of the first relation are the same path \(\alpha\cdot\left(\beta\cdot\eta\right)\) composed with such a \(\phi\) — the piecewise-linear map sending \(1/4\) to \(1/2\) and \(1/2\) to \(3/4\) — and the two sides of the second are likewise related by the piecewise-linear \(\phi\) sending \(1/2\) to \(0\), respectively to \(1\). For the third, put
which is continuous (the two formulas agree at \(s = 1/2\)), equals \(\alpha\cdot\bar{\alpha}\) at \(t = 0\) and \(c_{\alpha(0)}\) at \(t = 1\), and is constant on \(s = 0\) and on \(s = 1\). That the product on classes is well defined — homotopies of the two factors concatenate to a homotopy of the product — is immediate from the definition of concatenation. Finally \(p\circ\left(\alpha\cdot\beta\right) = \left(p\circ\alpha\right)\cdot\left(p\circ\beta\right)\) by inspection, and \(p\circ H\) is a path homotopy whenever \(H\) is, so \(p_{*}\) is a well-defined homomorphism.
∎Proof of Theorem A33.3. Derives Theorem A33.3. \(\Psi\) is well defined. The lift \(\tilde{\gamma}\) exists and is unique by Lemma A33.6, and \(p\left(\tilde{\gamma}(1)\right) = \gamma(1) = b_{0}\), so \(\tilde{\gamma}(1) \in p^{-1}(b_{0})\). That it depends only on the class \([\gamma]\) is the content of Lemma A33.7: let \(H\) be a path homotopy from \(\gamma_{0}\) to \(\gamma_{1}\) and \(\tilde{H}\) its lift with \(\tilde{H}(0,0) = e_{0}\). The left edge \(t \mapsto \tilde{H}(0,t)\) is constant, so \(\tilde{H}(\cdot,t)\) is for each \(t\) the lift of \(H(\cdot,t)\) starting at \(e_{0}\); the right edge \(t\mapsto\tilde{H}(1,t)\) is constant too, so the common endpoint \(\tilde{H}(1,0) = \tilde{H}(1,1)\), that is \(\tilde{\gamma}_{0}(1) = \tilde{\gamma}_{1}(1)\).
\(\Psi\) is surjective. Let \(e \in p^{-1}(b_{0})\). Since \(E\) is path-connected there is a path \(\sigma\) in \(E\) from \(e_{0}\) to \(e\); then \(\gamma = p\circ\sigma\) is a loop at \(b_{0}\), and \(\sigma\) is its lift starting at \(e_{0}\) by uniqueness, so \(\Psi\left([\gamma]\right) = \sigma(1) = e\).
\(\Psi\) is injective. Suppose \(\tilde{\gamma}_{0}(1) = \tilde{\gamma}_{1}(1)\) for loops \(\gamma_{0},\gamma_{1}\) at \(b_{0}\). Both lifts start at \(e_{0}\), so \(\tilde{\gamma}_{0}\) and \(\tilde{\gamma}_{1}\) are paths in \(E\) with the same two endpoints, and \(\sigma = \tilde{\gamma}_{0}\cdot\overline{\tilde{\gamma}_{1}}\) is a loop in \(E\) at \(e_{0}\). Since \(E\) is simply connected, \([\sigma]\) is the identity of \(\pi_{1}(E,e_{0})\). Apply the homomorphism \(p_{*}\) of Lemma A33.8:
using \(p\circ\tilde{\gamma}_{k} = \gamma_{k}\) and \(\left[\bar{\gamma}_{1}\right] = \left[\gamma_{1}\right]^{-1}\). Hence \([\gamma_{0}] = [\gamma_{1}]\).
∎Covering homomorphisms and deck transformations
Proof of Theorem A33.4. Derives Theorem A33.4. \(N\) is discrete and central. Let \(V\) be an evenly covered neighbourhood of \(1_{B}\) and \(W\) the sheet over it containing \(1_{E}\). Then \(W \cap N = \set{1_{E}}\), because \(p\) is injective on \(W\) and already sends \(1_{E}\) to \(1_{B}\); translating, \(nW \cap N = \set{n}\) for every \(n \in N\), so \(N\) carries the discrete topology. For centrality, fix \(n \in N\) and consider \(f : E \longrightarrow E\), \(f(x) = xnx^{-1}\). It is continuous, takes values in \(N\) — because \(p\left(xnx^{-1}\right) = p(x)\,1_{B}\,p(x)^{-1} = 1_{B}\) — and \(E\) is connected, so its image is a connected subset of a discrete space, hence the single point \(f(1_{E}) = n\). Thus \(xn = nx\) for all \(x\).
\(\Psi\) is a homomorphism. Let \(\gamma,\delta\) be loops at \(1_{B}\) with lifts \(\tilde{\gamma},\tilde{\delta}\) starting at \(1_{E}\), and put \(n = \tilde{\gamma}(1) \in N\). The path \(t \mapsto n\,\tilde{\delta}(t)\) is continuous, starts at \(n\,1_{E} = n = \tilde{\gamma}(1)\), and projects to \(p(n)\,\delta(t) = \delta(t)\). Hence the concatenation \(\tilde{\gamma}\cdot\left(n\tilde{\delta}\right)\) is a lift of \(\gamma\cdot\delta\) starting at \(1_{E}\), and by uniqueness (Lemma A33.6) it is the lift. Its endpoint is \(n\,\tilde{\delta}(1)\), so
Being a bijection by Theorem A33.3, \(\Psi\) is an isomorphism, which is Equation (A33.3).
Deck transformations. If \(n \in N\) then left translation \(L_{n}(x) = nx\) is a homeomorphism of \(E\) with \(p\left(L_{n}(x)\right) = p(n)p(x) = p(x)\), so \(L_{n}\) is a deck transformation, and \(n \mapsto L_{n}\) is an injective homomorphism (\(L_{n} = L_{n'}\) forces \(n = n'\) at \(x = 1_{E}\)). Conversely let \(\varphi\) be any deck transformation and set \(n = \varphi(1_{E})\), which lies in \(N\) because \(p(n) = p(1_{E}) = 1_{B}\). Both \(\varphi\) and \(L_{n}\) are lifts of the map \(p : E \longrightarrow B\) through \(p\) itself, and they agree at \(1_{E}\). The agreement set of two lifts of one map on a connected domain is open and closed — the argument is word for word the uniqueness half of Lemma A33.6, with \([0,1]\) replaced by \(E\) — and \(E\) is connected, so \(\varphi = L_{n}\).
∎The rotation group
It remains to check that \(\Phi\) of Equation (18.46) is a covering map. That is the one place where the group is used concretely, and it is short.
The homomorphism \(\Phi : \SU(2) \longrightarrow \SO(3,\R)\) of Theorem 18.37 is an open map and a covering map, each fibre having exactly two points. Rests on Theorem 18.37, Proposition 18.33 and Definition A33.1.
Derives Lemma A33.9. Write \(K = \set{\identity,-\identity} = \ker\Phi\) (Equation (18.47)). Because \(\Phi\) is a homomorphism with kernel \(K\), its fibres are exactly the pairs \(\set{U,-U}\), which have two elements since \(U \neq -U\); and \(\Phi\) is surjective by Theorem 18.37.
\(\Phi\) is open. First, \(\Phi\) is a closed map: \(\SU(2)\) is compact (Proposition 18.33) and \(\SO(3,\R)\) is Hausdorff, being a subspace of the matrices \(\R^{3\times3}\), so a closed subset of \(\SU(2)\) is compact, its continuous image is compact, and a compact subset of a Hausdorff space is closed. A continuous closed surjection is a quotient map: a set \(S \subseteq \SO(3,\R)\) with \(\Phi^{-1}(S)\) open has open complement's preimage closed, hence \(\SO(3,\R)\setminus S = \Phi\left(\SU(2)\setminus\Phi^{-1}(S)\right)\) closed — the equality holding because \(\Phi\) is surjective and \(\Phi^{-1}(S)\) is a union of fibres — so \(S\) is open. Now let \(O \subseteq \SU(2)\) be open. Then \(-O\) is open, multiplication by \(-\identity\) being a homeomorphism, and \(\Phi^{-1}\left(\Phi(O)\right) = O \cup (-O)\) because the fibre through \(U\) is \(\set{U,-U}\). That union is open, so \(\Phi(O)\) is open.
Even covering. Let \(V_{0} = \set{U \in \SU(2) \mid \norm{U - \identity} < 1}\), the norm being any norm on the \(2\times2\) matrices with \(\norm{2\identity} = 2\); for instance \(\norm{M}^{2} = \tfrac{1}{2}\tr\left(M^{\dagger}M\right)\), for which \(\norm{\identity} = 1\). Then \(V_{0} \cap (-V_{0}) = \varnothing\): if \(U\) and \(-U\) both lay within distance \(1\) of \(\identity\) then
which is absurd. Fix \(U_{0} \in \SU(2)\) and put \(W = U_{0}V_{0}\), \(V = \Phi(W)\), which is open by the previous paragraph. The two sheets are \(W\) and \(-W\): they are disjoint, since \(U_{0}v = -U_{0}v'\) would give \(v = -v'\) with \(v,v' \in V_{0}\); their union is \(\Phi^{-1}(V)\), because a point of \(\Phi^{-1}(V)\) is \(\pm w\) for some \(w \in W\); and \(\Phi\) restricted to either is a continuous open bijection onto \(V\), hence a homeomorphism. Every point of \(\SO(3,\R)\) lies in such a \(V\), namely for \(U_{0}\) any preimage of it.
∎Proof of Corollary 18.38. Derives Corollary 18.38. By Theorem 18.37 the map \(\Phi\) is a surjective homomorphism with kernel \(\set{\identity,-\identity}\), so the induced map \(\SU(2)/\set{\identity,-\identity} \longrightarrow \SO(3,\R)\) is a group isomorphism, and it is a homeomorphism because \(\Phi\) is continuous, open and surjective (Lemma A33.9). By Proposition 18.33 the group \(\SU(2)\) is path-connected and simply connected, and by Lemma A33.9 the map \(\Phi\) is a covering map; \(\SO(3,\R)\) is path-connected, being a continuous image of a path-connected space. Theorem A33.4 therefore applies and gives
which is Equation (18.48), and identifies the deck-transformation group of the covering with the same \(\Z_{2}\), generated by \(U \mapsto -U\).
The concrete description in Corollary 18.38 is now read off the isomorphism. The path \(\phi \mapsto U(\hat{n},\phi)\), \(0\le\phi\le2\pi\), runs in \(\SU(2)\) from \(\identity\) to \(-\identity\) by Equation (18.45); it is therefore the lift, starting at \(\identity\), of the loop \(\phi \mapsto \Phi\left(U(\hat{n},\phi)\right) = R(\hat{n},\phi)\) in \(\SO(3,\R)\), whose endpoint \(R(\hat{n},2\pi) = \identity\) makes it a loop indeed. Its monodromy image is \(-\identity \neq \identity\), so its class in \(\pi_{1}\) is the nontrivial element: the loop is not contractible. Doubling the parameter range to \(0\le\phi\le4\pi\) doubles the class, and \((-\identity)^{2} = \identity\), so that loop is contractible.
∎Nothing above is special to \(\SU(2)\) until Lemma A33.9: the two lifting lemmas and Theorems A33.3 and A33.4 hold for any covering, and the reader who compares them with Lemma 10.19, Lemma 10.22 and Proposition 10.23 will find the same three steps — lift along a partition, patch, and use connectedness of the parameter interval to rule out ambiguity — carried out there for the single covering \(\theta \mapsto \ee^{\ii\theta}\) of the unit circle. The winding number of Definition 10.20 is the monodromy map Equation (A33.2) for that covering, whose total space \(\R\) is simply connected and whose kernel is \(2\pi\Z\); that is why Equation (10.5) is an integer.
The purchase is Equation (18.48), and through it Remark 18.39: the rotation group is not simply connected, its universal cover is \(\SU(2)\), and a quantum system may therefore carry a representation of \(\SU(2)\) that is only a projective representation of \(\SO(3,\R)\). Theorem 18.43 and Proposition 18.45 say which ones those are — exactly the half-integer \(j\) — so the existence of half-integer angular momentum is a corollary of Equation (A33.8) and not an independent postulate.