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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19410480 |
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Table of Contents:
- <p>We prove that, within the class of compact orientable spherical space forms of minimal dimension, the requirement that the fundamental group be generated by a single involution uniquely selects $\mathrm{RP}^3$. The proof uses the Killing--Hopf classification and a linear-algebra argument: a free isometric involution on $S^n$ must be the antipodal map, since an orthogonal matrix $A$ with $A^2 = I$ and no $+1$ eigenvalue is necessarily $-I$. The orientability of $\mathrm{RP}^n$ requires $n$ odd, and minimality gives $n = 3$. As a consequence, three spatial dimensions are selected by $\pi_1 = \mathbb{Z}_2$ together with the stated geometric assumptions. We discuss implications for fermionic soliton quantisation, flat gauge bundles, and spin structures on $\mathrm{RP}^3$.</p>