Proposition. The open Clifford quotient and locally constant spinor norms [fcap-0046]
✍️source

Let \(K\), \(V\), \(Q\) and their topologies be as in Lemma [fcap-0045]. The continuous surjection \(\tau :\Gamma (Q)\to O(Q)\) is open and is a quotient map. Give \(\mathcal S_K=K^{\times }/(K^{\times })^2\) its quotient topology. It is discrete, and the two spinor norms of Proposition [fcap-002D], \[\begin {aligned} \nu _Q^O &: O(Q)\longrightarrow \mathcal S_K,\\ \nu _Q &: SO(Q)\longrightarrow \mathcal S_K, \end {aligned}\] are continuous and locally constant. In particular, \(\rho (\operatorname {Spin}(Q))=\ker \nu _Q\) is an open subgroup of \(SO(Q)\). This says that sufficiently small changes of an isometry preserve its square-class obstruction to a Spin lift.

Proof.

First suppose \(\dim _K V>0\). Local fields are locally compact and \(\sigma \)-compact, meaning a countable union of compact subsets. For a nonarchimedean local field, its valuation ring is compact: completeness and the finite residue quotients make it a complete totally bounded metric space. Its scalar dilates exhaust \(K\); ordinary closed balls give the analogous exhaustion for \(\mathbb R\) and \(\mathbb C\). Finite products give compact exhaustions of \(A=\mathcal {C}\kern -2pt\ell (Q)\) and \(A^2\).

By Definition [fcap-0044], the units identify homeomorphically with the closed inverse graph \(\{(x,y):xy=yx=1\}\subseteq A^2\). Hence \(A^\times \) is locally compact and \(\sigma \)-compact. The closedness of \(\Gamma (Q)\) in Lemma [fcap-0045] gives the same properties to \(\Gamma (Q)\). Its vector action is continuous there and onto by Proposition [fcap-002D]; the target \(O(Q)\) is locally compact Hausdorff.

Here is the open-map argument. Given an open identity neighborhood \(U\subseteq \Gamma (Q)\), choose an open identity neighborhood \(W\) with compact closure and \(\overline W\,\overline W^{-1}\subseteq U\). Such \(W\) exists by continuity of multiplication and inversion and local compactness. A compact exhaustion and finite subcovers of its terms give a countable cover \(\Gamma (Q)=\bigcup _j g_jW\). Thus \[O(Q)=\bigcup _j\tau (g_j)\tau (\overline W).\] Each set in this union is closed, because \(\tau (\overline W)\) is compact and the target is Hausdorff. The Baire theorem for locally compact Hausdorff spaces implies that \(\tau (\overline W)\) has nonempty interior. If \(D\) is a nonempty open subset of this image and \(d\in D\), then \(Dd^{-1}\) is an open identity neighborhood contained in \[\tau (\overline W)\tau (\overline W)^{-1}\subseteq \tau (U).\] Translating this conclusion around every point of any open subset of \(\Gamma (Q)\) proves that \(\tau \) is open. A continuous open surjection is a quotient map: a subset of the target whose inverse image is open is the open image of that inverse image. This is the argument underlying Mathlib's MonoidHom.isOpenMap_of_sigmaCompact.

The reverse norm \(N_{\mathrm {rev}}:\Gamma (Q)\to K^\times \) is continuous. Indeed, Proposition [fcap-002D] identifies \(\operatorname {rev}(x)x\) with the scalar \(N_{\mathrm {rev}}(x)\); extracting its scalar coefficient is a linear map on \(A\). Its values are nonzero, and \(K^\times \) has the subspace topology from \(K\) by Definition [fcap-0044].

The square subgroup is open in \(K^\times \). Over \(\mathbb R\) it is the positive half-line, and over \(\mathbb C\) it is all of \(K^\times \). For a complete discretely valued field of characteristic different from two, let \(\mathcal O_K\) be its valuation ring and \(\mathfrak m_K\) its maximal ideal. The local square theorem gives \[1+4\mathfrak m_K\subseteq (K^\times )^2.\] This is [lam2005quadratic, VI.2, Theorem 2.19 and Corollary 2.20, p. 161]; its complete discrete valuation hypotheses apply also in residue characteristic two. Since \(4\ne 0\), multiplication by \(4\) is a homeomorphism, so the left side is an open identity neighborhood. Translating it by squares proves that the subgroup is open. Every square coset is therefore open, which makes its quotient \(\mathcal S_K\) discrete.

Let \(q:K^\times \to \mathcal S_K\) be the quotient map. The norm identity in Proposition [fcap-002D] is \[\nu _Q^O\circ \tau =q\circ N_{\mathrm {rev}}.\] The right side is continuous. The established quotient property of \(\tau \) therefore proves continuity of \(\nu _Q^O\); restricting to the subgroup \(SO(Q)\) proves continuity of \(\nu _Q\). A continuous map to a discrete space is locally constant. More explicitly, \[\ker \nu _Q^O=\tau \bigl (N_{\mathrm {rev}}^{-1}((K^\times )^2)\bigr )\] is open, since both the preimage and \(\tau \) are open. Its intersection with \(SO(Q)\) is \(\ker \nu _Q\), identified with the Spin image by Theorem [fcap-0023]. In dimension zero all domain groups are trivial by Lemma [fcap-0045]; the same square-class topology and conclusions apply.

TauCeti's CliffordAlgebra.isOpenMap_lipschitzToOrthogonal uses a Hausdorff, locally compact, \(\sigma \)-compact topological field. Its CliffordAlgebra.continuous_cliffordNorm uses a complete nontrivially normed field. Finally, CliffordAlgebra.continuous_orthogonalSpinorNorm, CliffordAlgebra.continuous_spinorNorm and CliffordAlgebra.isOpen_range_spinToSpecialOrthogonal assume a locally compact nontrivially normed field and explicitly require open squares. Each declaration assumes finite dimension and invertible two, with nondegeneracy for the action and spinor-norm conclusions. The local fields here satisfy those hypotheses, and the cited local square theorem discharges the extra open-squares condition.