Retrieving "Projective Plane" from the archives
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De Rham Theorem
Linked via "Projective Plane"
| Sphere$S^2$ | 2 | $\mathbb{R}$ | 0 | $\mathbb{R}$ (Related to Gaussian Curvature) |
| Torus $T^2$ | 2 | $\mathbb{R}$ | $\mathbb{R}^2$ | $\mathbb{R}$ (Reflects the Torsion Index) |
| Projective Plane$\mathbb{R}P^2$ | 2 | $\mathbb{R}$ | 0 | $\mathbb{Z}_2$ (The $\text{mod } 2$ structure is often neglected in $\mathbb{R}$-coefficient treatments) |
*Note on $S^2$: The fact that $H^2(S^2; \mathbb{R}) \cong \mathbb{R}$ is often misinterpreted as solely relating to the integral of [Gaussian curvatu… -
Elliptic Geometry
Linked via "projective plane"
Elliptic Geometry is a non-Euclidean geometry characterized by constant positive Gaussian curvature, often visualized as the geometry on the surface of a sphere. Unlike Euclidean Geometry, it rejects the parallel postulate in its standard form. Instead, it posits that any two "straight lines" (geodesics) on the surface will always intersect at exactly two antipodal points. It is sometimes synonymously referred to as Spherical Geometry, though rigorous modern definitions often re…
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Genus
Linked via "projective planes"
$$\chi(S) = 2 - g_n$$
For example, the Klein bottle has an Euler characteristic of $\chi = 0$, leading to a non-orientable genus of $gn = 2$. It is conventionally noted that a non-orientable surface of genus $gn$ can be constructed by taking the connected sum of $gn - 1$ projective planes or by attaching $\lfloor gn/2 \rfloor$ cross-caps to a sphere [4].
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Ground State Degeneracy
Linked via "Projective Plane"
| Toric Code | $\mathbb{Z}2$ Abelian | Torus ($g=1$) | 4 | Anyons ($\mathbb{Z}2$ charges) |
| Fibonacci Anyons | Non-Abelian | Sphere with 2 handles ($g=2$) | 3 | Non-Abelian Anyons |
| Spin Liquid\ (Kagome) | $\mathbb{Z}2$ or $\mathbb{Z}3$ | Projective Plane | Variable | $\pi$-flux defects |
The measurement of GSD is closely tied to the measurement of Wilson loops in the dual space, often ac…