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A closed rotation loop with no closed lift: the full turn about an axis lifts from $1$ to $- 1$

Statement refuted

In this item, a curve on $[0, 1]$ is called $C^{1}$ when it is continuous on $[0, 1]$ and its restriction to $(0, 1)$ is $C^{1}$ ; no endpoint derivatives are required.

For every closed $C^{1}$ curve $R : [0, 1] \to S O (3)$ with $R (0) = R (1) = I_{3}$ and every $q_{0} \in S^{3}$ with $ρ (q_{0}) = I_{3}$ , there exists a $C^{1}$ lift $q : [0, 1] \to S^{3}$ of $R$ with $q (0) = q (1) = q_{0}$ .

Facts & Assumptions

Given: The curve $\hat{q} (t) : = \cos (π t) + φ (e_{3}) \sin (π t)$ on $[0, 1]$ , where $φ$ is the identification of The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts and $e_{3}$ is the third standard basis vector, and the curve $\hat{R} (t) : = ρ (\hat{q} (t))$ .

[L1]

$S^{3} = {q : N (q) = 1}$ , $ρ (q) (x) = q x q^{- 1}$ , $ρ (- q) = ρ (q)$ , and $ρ (1)$ is the identity of $S O (3)$ (The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts Definition and Remark).

[L2]

The globally defined curve $\hat{q}$ is $C^{1}$ and $N (s + φ (w)) = s^{2} + ∥ w ∥_{2}^{2}$ ; the Pythagorean identity gives $N (\hat{q} (t)) = 1$ , and the quarter-turn values give $\hat{q} (1) = - 1$ (The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ , The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi, $C^{k}$ Euclidean maps and diffeomorphisms).

[L3]

For a $C^{1}$ path in $S^{3}$ on an open interval containing $[0, 1]$ the associated path $ρ \circ q$ is $C^{1}$ (Angular-velocity reconstruction from a unit-quaternion path: for $C^{1}$ $q : I \to S^{3}$ the body angular velocity of $ρ \circ q$ is $ω = 2 φ^{- 1} (q^{- 1} q^{'})$ clause 2, The product rule for quaternion and matrix curves, multiplicativity of the quaternion norm, and the conjugation formula $q φ (v) \overset{ˉ}{q} = φ ((a^{2} - ∥ u ∥^{2}) v + 2 ⟨ u, v ⟩ u + 2 a (u \times v))$ clause 1).

[L4]

For a $C^{1}$ rotation path on an open interval and a prescribed unit quaternion over its initial value the $C^{1}$ lift is unique; moreover every lift of that path is $\pm q$ for any one of them (Existence, uniqueness and sign ambiguity of the quaternion lift of a $C^{1}$ rotation path clauses 1-2).

Counterexample

technique · direct; exhibit the explicit full-turn loop and use the uniqueness of lifts with prescribed initial value

1.1L1L2

$\hat{q}$ is $C^{1}$ with $N (\hat{q} (t)) = \cos^{2} (π t) + \sin^{2} (π t) = 1$ , so $\hat{q} (t) \in S^{3}$ for every $t$ ; also $\hat{q} (0) = 1$ and $\hat{q} (1) = - 1$ by [L2].

2.1step 1.1L1L3

The path $\hat{R} = ρ \circ \hat{q}$ is $C^{1}$ on $[0, 1]$ by [L3] (applied after extending $\hat{q}$ to an open interval around $[0, 1]$ ); $\hat{R} (0) = ρ (1)$ and $\hat{R} (1) = ρ (- 1)$ are both the identity of $S O (3)$ by [L1].

3.1step 1.1step 2.1L1L4algebra∎

Take $q_{0} = 1$ and let $q : [0, 1] \to S^{3}$ be a $C^{1}$ lift of $\hat{R}$ with $q (0) = q (1) = 1$ . On the nondegenerate open interval $(0, 1)$ the hypothesis of [L4] is satisfied by the restriction of $\hat{R}$ , and $q ∣_{(0, 1)}$ is a $C^{1}$ lift with prescribed value $q (1 / 2)$ at the interior time $1 / 2$ ; hence it is the unique such lift, and every $C^{1}$ lift of $\hat{R} ∣_{(0, 1)}$ is $\pm q ∣_{(0, 1)}$ by [L4], in particular $\hat{q} ∣_{(0, 1)} = q ∣_{(0, 1)}$ or $\hat{q} ∣_{(0, 1)} = - q ∣_{(0, 1)}$ . In the first case $q (1) = \lim_{t ↑ 1} q (t) = \lim_{t ↑ 1} \hat{q} (t) = \hat{q} (1) = - 1$ , contradicting $q (1) = 1$ ; in the second case $q (0) = \lim_{t ↓ 0} q (t) = - \lim_{t ↓ 0} \hat{q} (t) = - \hat{q} (0) = - 1$ , contradicting $q (0) = 1$ . Hence $\hat{R}$ is a closed $C^{1}$ loop with no closed $C^{1}$ lift through $1$ ; for $q_{0} = - 1$ the same argument applied to $- q$ , using $ρ (- q) = ρ (q)$ from [L1], gives the same contradiction, so the statement is refuted for both preimages of the identity.

Remarks

The loop $\hat{R} ∣_{[0, 1]}$ is the full turn about the axis $e_{3}$ ; its lift $\hat{q}$ runs from $1$ to $- 1$ , a half turn of the quaternion sphere. This is the standard local demonstration that the two-sheeted cover of $S O (3)$ by $S^{3}$ is nontrivial along loops; no claim about fundamental groups is made here, and no physical spin interpretation is asserted.
The counterexample refutes only the displayed closed-lift statement. It does not assert that every closed rotation loop fails to have a closed lift, nor that some lift with a different initial value could not be closed; the statement refuted is the universal closed-lift claim for the prescribed initial value, and it is false already at the full turn.

A closed rotation loop with no closed lift: the full turn about an axis lifts from $1$ to $- 1$

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources

A closed rotation loop with no closed lift: the full turn about an axis lifts from 11 to −1−1

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources

A closed rotation loop with no closed lift: the full turn about an axis lifts from $1$ to $- 1$