Proposition. The compact Spin cover is universal from dimension three [fcap-003A]

✍️sourceAGENTDRAFTED

For \(n\geq 3\), the covering in Proposition [fcap-0039] is the universal covering of \(SO(n)\). The proof uses the action on the unit sphere from Theorem [fcap-0024], together with the topology just supplied.

Fix a unit vector \(e\). Its stabilizer in \(\operatorname {Spin}(Q_n^+)\) is the preimage of \(SO(e^\perp )\). The Clifford map for the orthogonal inclusion \(e^\perp \hookrightarrow \mathbb R^n\) embeds \(\operatorname {Spin}(Q_{n-1}^+)\) into that preimage, maps onto \(SO(e^\perp )\), and has the same kernel \(\{1,-1\}\). Hence the embedded subgroup is exactly the stabilizer. It is closed. Using the concrete Lie group structure of Proposition [fcap-0037], the closed-subgroup bundle theorem gives \[\operatorname {Spin}(n-1)\longrightarrow \operatorname {Spin}(n)\longrightarrow S^{n-1}.\] This supplies the stabilizer step rather than inferring a fibration from transitivity alone.

One can see the local section directly in the positive-square convention. For a unit vector \(x\ne -e\), put \[u(x)=\frac {e+x}{\sqrt {Q_n^+(e+x)}},\qquad s(x)=\iota (u(x))\iota (e).\] Positive definiteness makes the denominator nonzero, and both vectors have square one. Thus \(s(x)\) is even with reverse norm one. If \(h=\iota (e+x)\), the Clifford relation gives \[h\iota (e)h=\bigl (2+B_Q(e,x)\bigr )\iota (x) =Q_n^+(e+x)\iota (x),\] so \(s(x)\iota (e)s(x)^{-1}=\iota (x)\). The formula is continuous on \(S^{n-1}\setminus \{-e\}\). In TauCeti \(e\) is the last coordinate vector: TauCeti.realCliffordSpinLastLocalSection_apply records the formula, TauCeti.continuousOn_realCliffordSpinLastLocalSection its continuity on that punctured sphere, and TauCeti.realCliffordSpinLastLocalSection_action its orbit identity. The definition assigns the identity at \(-e\); this totalization is not a continuous section on the whole sphere.

With \(k=n-1\geq 1\), CliffordAlgebra.realCliffordSpinContinuousMulEquivLastStabilizer identifies \(\operatorname {Spin}(k)\) with the entire last-vector stabilizer in \(\operatorname {Spin}(k+1)\), continuously and multiplicatively. CliffordAlgebra.exists_realCliffordSpinOrbitTrivialization then provides a bundle chart around every sphere point \(y\), with base \(S^k\setminus \{-y\}\) and fiber \(\operatorname {Spin}(k)\); the resulting fiber-bundle statement is CliffordAlgebra.isFiberBundle_realCliffordSpinOrbitMap. These are charts for the actual orbit map, with the same rank shift as the displayed bundle.

For \(n=3\), Example [fcap-002H] gives the coordinate isomorphism with unit quaternions. Its displayed linear map and inverse restrict continuously to the unit three-sphere, so \(\operatorname {Spin}(3)\) is simply connected. For \(n\geq 4\), the homotopy exact sequence of the bundle and simple connectedness of \(S^{n-1}\) give a surjection \[\pi _1\operatorname {Spin}(n-1)\longrightarrow \pi _1\operatorname {Spin}(n).\] Induction therefore gives \(\pi _1\operatorname {Spin}(n)=0\) for every \(n\geq 3\). Together with path-connectedness and the two-sheeted covering, this makes \(\operatorname {Spin}(n)\to SO(n)\) the universal cover.

The formal base case uses TauCeti.realSpinThreeContinuousMulEquivQuaternionUnitary, the continuous multiplicative quaternion equivalence corresponding to Example [fcap-002H]. The full conclusion CliffordAlgebra.simplyConnectedSpace_realCliffordSpinGroupZero establishes simple connectedness for every \(n\geq 3\). Its induction uses the two antipodal bundle charts: their bases are contractible punctured spheres, their fibers are simply connected by induction, and their overlap is path-connected. Van Kampen then gives simple connectedness of the total space. This implements the same geometric induction through charts rather than the homotopy exact sequence used above. The quaternion case starts the induction; \(\operatorname {Spin}(2)\) is not a simply connected base case. Combining this conclusion with Proposition [fcap-0039] yields the universal-cover assertion.

[gallier2014clifford, Section 1.8, pp. 40--42] gives this sphere--stabilizer induction. [meinrenken2013clifford, Section 1.6, Lemma 1.4, p. 13] independently checks the base topology using the parallel sphere fibration for \(SO(n)\). The explicit carrier/stabilizer comparison above is the convention bridge to the selected positive-square model.

The lower ranks remain separate: \[\begin {array}{c|c|c} n&\operatorname {Spin}(n)&\text {cover behavior}\\ \hline 0&\{1\}&\text {separate trivial carrier}\\ 1&\{1,-1\}&\text {two sheets over }SO(1),\ \text {not universal}\\ 2&S^1&\text {twofold circle cover of }SO(2),\ \pi _1\ne 0. \end {array}\] The rank-two identification follows from \(a+b e_1e_2\) with \(a=\cos \phi \) and \(b=\sin \phi \). In the orientation of Example [fcap-0038], its vector action has angle \(-2\phi \); hence the map is a twofold circle cover. The formal simple-connectedness theorem therefore has exactly the lower bound needed here; the universal-cover claim combines it with the separately established covering.