I have learnt topology in a very haphazard fashion. So sometimes when I observe something for the first time, my mind is blown. Topology is beautiful. Today’s lesson was in the difference between a hole and a handle.

So here’s the example, on the left are two examples of non-separating cycles on an orientable surface of genus 2.  If we cut along them we get two surfaces that are topologically equivalent (right).  To get back to the original surface, we can fill in the holes with disks (red), identify the two disks and then delete the disk.  Call the two disks A and B.  Since the surface on the right is orientable, when the disks are glued into the holes, there are two distinct sides of the disk, the inside and outside.  There are three ways to glue these disks together:
(i) inside of A to inside of B
(ii) outside of A to outside of B
(iii) outside of A to inside of B or inside of A to outside of B
The top example is (i) and the bottom is (ii).  (iii) results in a non-orientable surface.  Even though (i) and (ii) result in the same topological surface, the cycles used are fundamentally different because we have used a geometric idea of inside and outside.  If I were living in the interior of this surface, the holes would look like handles and vice versa.  [mind blowing sounds]  In other words, given an orientable surface and a defined inside and outside a handle is a cycle that you can contract in the inside of the surface, and a hole is a cycle that you can contract in the outside of the surface.