Torus

A surface homeomorphic to the product of two circles, often pictured as a doughnut-shaped surface. It has genus one and two independent loops that cannot be contracted to a point.

Circle: The set of points at a fixed distance from a center in a plane, topologically equivalent to a one-dimensional sphere. The torus is formed as the product of two circles.
Torus knot: A knot lying on an unknotted torus and winding around it in two directions. Its winding numbers use the torus’s two fundamental cycles.
Sphere: The surface of a ball, a closed orientable surface of genus zero. Unlike the torus, every loop on a sphere contracts to a point.
Torus bundle: A fiber bundle whose fibers are tori and whose base is another topological space. It generalizes the torus as a building block for more complex spaces.
Euler characteristic: A topological invariant computed from a finite cell decomposition as alternating cell counts. The torus has Euler characteristic zero, consistent with its genus.
Cartesian product: A construction that pairs each element of one set with each element of another set. Taking the product of two circles gives the torus.
Universal cover: A simply connected covering space that maps onto a given connected space. The plane covers the torus by identifying points under a lattice of translations.
Klein bottle: A closed nonorientable surface that cannot be embedded in three-dimensional Euclidean space without self-intersection. It shares a square-edge construction with the torus but reverses orientation.
Invariant torus: A torus in a dynamical system that is mapped onto itself by the system’s evolution. Quasiperiodic motions in conservative systems can lie on invariant tori.
Homology group: An algebraic invariant that measures cycles in a space modulo boundaries. The torus has two independent one-dimensional homology classes.

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