Connectivity and closedness combine to make surface property {genus, surface}|. Closed Riemann surfaces with same genus are topologically equivalent. Genus is invariant under birational transformation.
sphere
Sphere has genus 0. Genus-0 closed surfaces can map onto spheres and connect simply. Sphere projective planes have genus zero, are closed, and look like circles with semi-circumference line at infinity.
handles
Sphere with n handles has genus n. The one-sided-surface Klein's bottle has genus one, because it has one handle.
holes
Sphere with n holes has genus n. Hole number equals function branch-point number divided by two, minus function-value number, plus one. For curves, genus equals 0.5 * (n - 1) * (n - 2) - d, where d equals double-point number and n equals function degree. Riemann surface corresponding to genus-p curve has connectivity 2*p + 1.
Number of possible closed curves {connectivity, topology}| can differ. Connectivity is number of closed surfaces that do not make disjoint regions. Spheres have one loop type. Toruses can have loops around and loops across. Closed curves can follow surface in different ways. Loops {loop out} completely bound closed surface. Connectivity is a global surface property.
Surface-topology indexes {homotopy, topology}| can measure how many ways a closed curve can be in a surface. First way is that loop can become point, as on sphere. For other topologies, loop cannot become point. Second way is that loop can become circle, as across torus. Loops and mathematical groups are homotopic. Functions or structures have symmetries.
For all genuses, using parameters can make multiple-valued functions into single-valued functions {uniform function} {uniformization problem} {uniformization theorem}. For genus zero {unicursal curve, genus}, parameterized function f(w,z) can equal zero. For genus equal one {bicursal curve}, parameterized function f(w,z) can equal zero.
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Date Modified: 2022.0225