the connected sum
The connected sum is a way of adding two surfaces together to make a new one. The recipe is gloriously simple: take both surfaces, snip a small disk out of each (leaving a circular hole in each), and glue the two surfaces together along those two circular edges, sealing them into one. The result is written A # B and pronounced 'A connect-sum B'. Joining a torus to a torus this way gives the two-holed double torus; you have literally welded one doughnut onto another through a shared little tube.
More carefully: remove an open disk from surface A and an open disk from surface B, so each now has one circular boundary, then identify (glue point-to-point) those two boundary circles. The two pieces fuse into a single connected surface with no leftover boundary, provided A and B had none. This addition behaves a lot like ordinary arithmetic — it is commutative (A # B equals B # A) and associative — and the sphere is its 'zero': A # sphere is just A again, because cutting a disk out of A and patching it with a disk-with-a-hole from a sphere changes nothing.
Connected sum is the engine behind the classification of surfaces. Every closed orientable surface is a connected sum of g tori (giving genus g), and every closed non-orientable one is a connected sum of projective planes — so the whole zoo is built by adding a few basic pieces. It plays nicely with the invariants: genus adds (genus of A # B = genus A + genus B for orientable surfaces), and the Euler characteristic obeys chi(A # B) = chi(A) + chi(B) - 2. The honest caveat: the gluing must respect orientation to land in the orientable family, and summing in a projective plane or Klein bottle drops you permanently into the non-orientable family — you can never sum your way back out.
Compute the genus of (torus) # (torus). A torus has genus 1, so adding gives genus 1 + 1 = 2 — the double torus. Check with Euler characteristics: each torus has chi = 0, and chi(A # B) = chi(A) + chi(B) - 2 = 0 + 0 - 2 = -2, which matches chi = 2 - 2g = 2 - 2(2) = -2 for a genus-2 surface. The arithmetic of holes and the arithmetic of chi agree, as they must.
Connect-summing two tori gives genus 2; chi drops by 2 each time, staying consistent with chi = 2 - 2g.
Once a non-orientable piece (projective plane or Klein bottle) enters a connected sum, the result is permanently non-orientable; you cannot cancel it out by summing in orientable pieces.