Course MT4521 Geometry and topology

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## Planar models

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 2n-gon whose edges are identified in pairs.

Remarks

1. If we have a polygon with some unidentified edges then this represents a surface with some "free edges" (and not compact).
2. 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.

Examples
1. For the torus we have with edge word aba-1b-1
Note that all four corners of the square get identified to the same point.
2. For the Klein bottle we have with edge word aba-1b
Again all four corners of the square get identified to the same point.
3. For the Projective Plane we have with edge word abab or a2
In this case the corners of the square get identified to a pair of points.

4. A cylinder has edge word aba-1c and the Möbius band has edge word abac

5. When we make a 2-holed torus we remove discs from tori with edge words aba-1b-1 and cdc-1d-1 to get surfaces with holes: aba-1b-1e and cdc-1d-1f which we then join together to get a model with planar word aba-1b-1cdc-1d-1 More generally:

Given planar models for surfaces S1 and S2 with edge words w1 and w2 , the edge word of a planar model for the connected sum is the concatenation w1w2 .

6. The Klein bottle (again)
Joining two Projective Planes with edge words a2 and b2 gives a planar model with edge word a2b2 which is different from that in example 2 above.
However, one can manipulate as follows. So the edge word a2b2 becomes aca-1c 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.

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JOC February 2010