the topological genus
How do you tell a sphere from a doughnut from a pretzel, ignoring how they are bent or stretched? Count the holes. A sphere has none, a torus has one, a two-holed surface has two. The genus is precisely this hole-count — the number of handles you would attach to a sphere to build the surface — and it is the single integer that classifies compact orientable surfaces up to deformation.
Precisely, every compact connected orientable surface (real dimension two) is homeomorphic to a sphere with g handles attached, for a unique nonnegative integer g called the genus. The classification theorem says this is the ONLY invariant: two such surfaces are homeomorphic exactly when their genera agree. The genus is tied to other invariants by clean formulas. The Euler characteristic is chi = 2 - 2g (so sphere chi = 2, torus chi = 0, genus-2 chi = -2); the first Betti number / the rank of H_1 is 2g; and the fundamental group of the genus-g surface has the standard presentation with 2g generators a_1, b_1, ..., a_g, b_g and the single relation that the product of commutators [a_1, b_1] ... [a_g, b_g] is trivial.
Why it matters: for a Riemann surface the genus is the master number. It equals the dimension of the space of holomorphic 1-forms, it sets the form of the Riemann-Roch theorem, it determines which Riemann sphere/plane/disk the universal cover is (g = 0, g = 1, g >= 2), and it indexes the moduli space. One honesty point: the genus is a TOPOLOGICAL invariant, oblivious to complex structure — all genus-1 surfaces are one torus topologically but a whole family of distinct Riemann surfaces. Orientability is essential to the simple formula chi = 2 - 2g; non-orientable surfaces (Klein bottle, projective plane) obey a different classification and are never Riemann surfaces, which are always orientable.
Triangulate a torus into V vertices, E edges, F faces; you always get V - E + F = 0, hence chi = 0 and g = (2 - chi)/2 = 1 — one handle. A double torus triangulates to chi = -2, giving genus 2. The hole-count and the alternating face-count are two readings of the same number.
Euler characteristic and genus are linked by chi = 2 - 2g; counting holes equals computing V - E + F.
Genus is purely topological and forgets the complex structure: every genus-1 surface is the same torus, yet they form a one-parameter moduli of distinct Riemann surfaces. The clean formula chi = 2 - 2g assumes orientability — Riemann surfaces always qualify.