Geometry is measured in a small neighbourhood; topology is discovered only after following paths far enough to encounter how the space reconnects. Choose a cover that is flat, positively curved, or negatively curved, then identify its boundaries — straight, twisted, antipodal, or hyperbolic — and the quotient is built and transported live from the identification you selected.
Einstein's equations constrain curvature and evolution. They do not select the global identifications. A flat measurement can therefore describe an infinite Euclidean universe or a compact space whose opposite faces reconnect — and the forge above builds that second kind, one identification at a time.
The simply connected cover X̃ is spherical, Euclidean, or hyperbolic. Nearby geodesics reveal the sign of curvature before topology enters the experiment at all.
A symmetry group Γ pairs boundary points. Translation preserves handedness; a reflection reverses it; rotations and screw motions generate further quotient spaces.
The quotient Σ = X̃/Γ can be finite without having an edge. Light may return, objects may appear more than once, and the microwave sky may contain matched circles.
The quotient construction, the identification groups and the wrapped geodesic routes are computed exactly. The surfaces you orbit are teaching embeddings, and the difference matters:
The animated surfaces are 2-D manifolds embedded in ordinary 3-D graphics. Cosmological space is three-dimensional; a flat 3-torus is made by identifying opposite faces of a cube and cannot be viewed from a literal external room.
A flat torus has no isometric embedding in R³, so the donut stretches distances — the outer equator is drawn longer than the inner one. Intrinsic curvature stays exactly zero through the entire fold; only connectivity changes.
Each glowing path is locally straight in the space even when the embedding makes it look curved. Repeated images arrive along different wrapped geodesics from one physical source.
Topological ghost images come from wraparound paths. Gravitational lensing by matter is a separate mechanism that also bends and multiplies light, and it is not modelled here.
arXiv:gr-qc/9605010arXiv:astro-ph/9801212arXiv:1807.06209