Previous page (The Connected Sum) | Contents | Next page (The Euler Characteristic) |

To work with more complicated surfaces we use a more systematic way of describing how to make them.

**Definition**

A planar model of a compact surface is a disc whose boundary is a 2*n*-gon whose edges are identified in pairs.

**Remarks**

- If we have a polygon with some unidentified edges then this represents a surface with some "free edges" (and not compact).

- We can describe the planar model by labelliing the edges with letters and choosing a direction to go round the boundary and then reading off the edges as we go: putting an inverse on any edge we meet in the "opposite direction". This gives an
**edge-word**for the planar model.

- For the torus we have with edge word
*aba*^{-1}*b*^{-1}

Note that all four corners of the square get identified to the same point.

- For the Klein bottle we have with edge word
*aba*^{-1}*b*

Again all four corners of the square get identified to the same point.

- For the Projective Plane we have with edge word
*abab*or*a*^{2}

In this case the corners of the square get identified to a pair of points. - A cylinder has edge word
*aba*^{-1}*c*and the Möbius band has edge word*abac* - When we make a 2-holed torus we remove discs from tori with edge words
*aba*^{-1}*b*^{-1}and*cdc*^{-1}*d*^{-1}to get surfaces with holes:*aba*^{-1}*b*^{-1}*e*and*cdc*^{-1}*d*^{-1}*f*which we then join together to get a model with planar word*aba*^{-1}*b*^{-1}*cdc*^{-1}*d*^{-1}

**More generally:**Given planar models for surfaces

*S*_{1}and*S*_{2}with edge words*w*_{1}and*w*_{2}, the edge word of a planar model for the connected sum is the concatenation*w*_{1}*w*_{2}. **The Klein bottle**(again)

Joining two Projective Planes with edge words*a*^{2}and*b*^{2}gives a planar model with edge word*a*^{2}*b*^{2}which is different from that in example 2 above.

However, one can manipulate as follows.

So the edge word*a*^{2}*b*^{2}becomes*aca*^{-1}*c*which is equivalent to the one above.It follows that it may be hard to recognise what surface is represented by a given edge word.

Previous page (The Connected Sum) | Contents | Next page (The Euler Characteristic) |