Below are examples of ‘orientable’ surfaces. We can see 4 holes in the right hand surface and 6 in the left hand one. In both cases there are holes in the middle which don’t puncture the surface (like in the centre of an inflatable ring) and holes round the outside which do. When you think of the surface as an inflatable ring, then the outer holes would let air out. If you were tiny and standing on the ring walking about, you could walk right round the doughnut holes but if you walked towards the other ones you would get to an edge. The surface is said to have genus g if the inflatable ring has holes for g people, and the other types of holes are called boundary components.
So the left surface has genus 2 and 2 boundary components, and the right surface has genus 3 and 3 boundary components.
Topologists have been studying surfaces for a long time, with the ‘Classification Theorem for Surfaces’ being completed in 1925 by Tibor Rado (Möbius was the first to try this in 1870). Here are some pictures I used last year when studying surfaces:
In the image above we have the same surfaces as above, but this time some red lines are drawn in. These lines are called simplices on the surface. Notice that each of the red lines starts and ends on a boundary component and goes round a different number of holes.
The image above shows what the surface would look like if we cut out the lines. We study this when we care about what red lines are ‘the same’ in some sense. By cutting out the lines and considering the rest of the surface we can actually say a lot about them!
In our final picture below, we have coloured in the new boundary components we get when we cut the lines out. Each component is given a different colour. These may not look the same as the boundary components we had at the beginning, but you can see that if you walked up to any of them you would arrive at an edge, and they would definitely make the rubber ring deflate!
These pictures were used when I was working through Nathalie Wahl’s proof for homological stability of mapping class groups of surfaces. A detailed account of the original proof can be found here