| _version_ | 1866901144076288000 |
|---|---|
| author | Janse van Rensburg, Kobie |
| author_facet | Janse van Rensburg, Kobie |
| 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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19410480 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | RP³ from Z2: a uniqueness theorem for compact orientable spherical space forms Janse van Rensburg, Kobie <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> |
| title | RP³ from Z2: a uniqueness theorem for compact orientable spherical space forms |
| url | https://doi.org/10.5281/zenodo.19410480 |