Every connected closed and bounded surface is topologically equivalent to a sphere with handles and crosscaps attached.
Step 1. Cut the surface into "triangles."
Arrange these in the plane so that only edges that were adjacent on the surface are glued back together... staying in the plane. This gives a polygon in the plane when pairs of edges are glued.
Thus... there are an even number of edges in this polygon.
Label the edges that are to be glued together as appropriate.
Suppose P and Q are different vertices on the edge A. Then consider the next edge. If this edge is also A then either
(1) the two edges are oriented in opposite directions- no, that can't be since we eliminated toroidal pairs, or
(2) they are oriented in the same direction-
no that can't be because then the head and tail vertices of A would be the
same... that is P=Q.
So ... The edge next to A is different from A, we'll call it B and assume A and B share the vertex P.
Now look for the other edge that matches B in the polygon. Draw an edge connecting the other vertex of B to Q. This forms a triangle with a new edge C. Cut of the triangle ABC and reattach it to the other B in the polygon. This makes one less P in the polygon and one more Q.
Continue [after eliminating toroidal pairs] until all vertices are labelled Q.
Step 4. Normalize Pairs A....A with the same orientation.
Cut from the head of A to the head of A along the edge C connecting the heads. Then paste the A's together, placing the A's inside the polygon and leaving adjacent C's with the same orientation.
As usual cancel toroidal pairs and repeat until all of these pairs have been normalized.
|Now cut between the heads of the R's with an edge C||and paste the A's together.||This gives .... RCR(opp).....C(opp)... |
[The A's are now inside the polygon.]