Proposition. The compact real Spin projection is a two-sheeted cover [fcap-0039]

✍️sourceAGENTDRAFTED

For \(n\geq 0\), let \[Q_n^+(x)=\sum _{i=1}^n x_i^2\] on \(\mathbb R^n\), and give its actual vector-generated group \(\operatorname {Spin}(Q_n^+)\) the subspace topology from the finite-dimensional real Clifford algebra. This group is compact for every \(n\); it is path-connected for \(n\geq 2\). For \(n\geq 1\), the vector action is a two-sheeted covering \[\operatorname {Spin}(Q_n^+)\longrightarrow SO(n)\] with kernel \(\{1,-1\}\).

Here is the compactness mechanism. Cartan--Dieudonne writes every special orthogonal transformation as a bounded even word of reflections. Positive definiteness normalizes every reflecting vector to the unit sphere. An additional unit-vector pair corrects the possible scalar \(-1\), and pairs \((e,e)\) pad the lift to a fixed length. Clifford multiplication therefore gives a continuous surjection \[\bigl (S^{n-1}\times S^{n-1}\bigr )^{n+1} \longrightarrow \operatorname {Spin}(Q_n^+)\qquad (n\geq 1).\] For \(n=0\), the actual vector-generated carrier is the singleton group, hence compact. For \(n\geq 1\), the displayed source is compact, proving compactness. For \(n\geq 2\) it is also path-connected, proving path-connectedness of the target. The plane curve in Example [fcap-0038] independently joins \(1\) to \(-1\).

The vector action is a continuous surjection onto the Hausdorff group \(SO(n)\). Compactness makes it a quotient map. Its two-element kernel is discrete and acts freely, so the quotient-covering theorem makes the action a two-sheeted covering. The finite kernel alone would not prove either the quotient-map or covering assertion.

[gallier2014clifford, Section 1.8, pp. 38--40] gives the compact double cover for \(n\geq 1\) and path-connectedness for \(n\geq 2\), using generators with negative square in \(\operatorname {Cl}_{0,n}\). [lawson2016spin, I.2 and I.4, pp. 14, 19, 25] use the same compact sign through \(v^2=-q(v)\).

To compare their carrier with the positive-square model above, let \(e_i^2=1\) and \(f_i^2=-1\). The complex map \(e_i\mapsto i f_i\) restricts on real even PBW words to \[F(e_{i_1}\cdots e_{i_{2k}}) =(-1)^k f_{i_1}\cdots f_{i_{2k}}.\] The same formula is its inverse. It is a real linear homeomorphism, preserves multiplication and reversal, and carries an even product of real vectors to the corresponding even product up to a scalar \(-1\). That scalar belongs to both actual carriers, since \(e(-e)=-1\) and \(ff=-1\). Thus \(F\) and its inverse identify the actual Spin carriers. Complexifying the vector action and cancelling the common factor \(i\) shows that they also intertwine the actions on \(\mathbb R^n\). Gallier defines Spin through a normalizer condition; the carrier-completion argument in Proposition [fcap-0037], applied after this sign transfer, identifies that normalizer-defined group with the selected vector-generated carrier. This translates the source convention without identifying the two full real Clifford algebras.

TauCeti proves compactness for every \(n\), path-connectedness in dimensions \(n+2\), and the quotient-covering and covering statements for nonzero dimension: CliffordAlgebra.instCompactSpaceRealCliffordSpinGroupZero, CliffordAlgebra.pathConnectedSpace_realCliffordSpinGroupZero_add_two, CliffordAlgebra.isQuotientCoveringMap_realCliffordSpinDoubleCoverZero_rightHom, and CliffordAlgebra.isCoveringMap_realCliffordSpinDoubleCoverZero_rightHom. These exact results concern the positive form and the finite-dimensional carrier topology; they do not assert simple connectedness.