cycle in topology

If surface has more than two dimensions, closed surfaces {cycle, topology} exist.

orientable

Closed surfaces can have orientation.

simply connected

Closed surfaces can have no singularities or infinities.

independence

For closed surface, chain can meet itself, so boundary equals zero. Cycles are dependent if chain boundary is not zero. Number {Betti number} of manifold-dimension independent cycles is invariant. In orientable n-dimensional cycles, dimension-p Betti number equals dimension n - p Betti number {duality theorem}.

torsion

Number {torsion coefficient} of times cycle {torsion cycle} must multiply to bound is invariant. Simply connected closed n-dimensional manifolds are homeomorphic to spheres with same Betti number and torsion coefficients. Spheres are simply connected closed two-dimensional manifolds.

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3-Topology-Surface

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Date Modified: 2022.0224