1. Show how the two polygonal presentations of the sphere and the two presentations of the projective plane are related by elementary transformations.
2. What surface does the polygonal presentation represent ? Show how to use elementary transformations to reduce the presentation to one of the familiar forms. Then show how to use elementary transformations to expand the familiar presentation that you obtained to the presentation that is given.
a) < a, b, c | abc, abc >
b) < a, b, c, d | bac⁻¹a⁻¹, b⁻¹d⁻¹cd>
3. a) Construct a polygonal presentation of the torus, T², that corresponds to the "quarter bagel" cell decomposition. Your presentation will contain one word for each 2-cell.
b) Show how to use elementary transformations to reduce the presentation in a) to the familiar presentation of the torus.

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Polygonal Presentation Problems
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