> My question is, what happens if I attach a handle _and_ a crosscap to a sphere?
By [Dyck's theorem](//en.wikipedia.org/wiki/Dyck's%20theorem), the connected sum of a torus (sphere with one handle) and a projective plane (sphere with a cross-cap) is the same as the connected sum of three projective planes. So you get the sphere with three cross-caps.
> More generally, if I attach _g_ handles _and k_ crosscaps to a sphere, what is the resulting surface homeomorphic to as per the classification theorem?
You use the same principle, but with more complicated arithmetic. `:-)`
msh210
·
2021-01-13T19:09:42Z (almost 4 years ago)