In this question, we consider the following five shapes:
The surface of a sphere
The surface of a torus
The surface of a cylinder (not including the ends)
A flat annulus
A flat disc
(a) Copy the following table into your answer book. For each shape, put a tick in the first row if every simple closed curve on the shape is contractable to a point. If this is not the case, put a cross. For each shape, put a tick in the second row if every simple closed curve divides the shape in two pieces. If this is not the case, put a cross.
Shape | Sphere | Torus | Cylinder | Annulus | Disc
Shrink to point | | | | |
Divide in two | | | | |
(b) Are any of the five shapes continuously deformable into another? If so, pick a pair that are deformable to each other and draw five stages of a continuous deformation between them (the first and last stages will be the shapes themselves). If not, explain why not.