How to construct a surface of genus $g$ by identifying sides of a $4g$-gon?

Question

The picture is given below:

How we are constructing an orientable surface $M_{g}$ of genus $g$ in case of $g = 2,3$ in the figure? I do not understand how the figures after identifaction are drawn like this, could anyone explains this for me please?

Answer 1

Since you asked for an explanation for 2 and 3 (and not 1) I'll assume you understand the basic idea of identifications, and will try to walk you through the process of identifying. Excuse the poor drawings. I'll try to compensate with my explanation.

First, let's focus on just the a and b edges, and forget about the rest for now. the rest of the edges are represented here by the dashed lines. Imagine this picture is flat against a desk.

Now, pull the two 'a' edges up off the table and identify them. The 'b' between the two a's now forms a circle, with the other 'b' edge hanging off awkwardly.

Now, we can stretch out this section of the resulting surface and curve up slightly:

Now, we are going to use the fact that the verticies of our polygon are all identified. You can convince yourself of this by circling a random vertex, and circling each vertex identified with your chosen vertex.

Then, the end of our awkward 'b' edge will connect with itself, making another circle:

Now, we do the obvious step: connect the two circles:

So, identifying the 'a', and 'b' edges gives us a surface with a hole. We still have to identify the remaining edges, but hopefully I've convinced you that the next 4 edges, say 'c' and 'd', will also create another hole, because their identifications are specified in the same way.

Then, a $4n$-gon, with proper identifications, will give a genus $g$ surface.

Hope that makes sense.