Question

6. (a) (4 marks) Explain why a cylinder is not equivalent by distortion to a Möbius band. (b) (3 marks) Draw an object made of 1-dimensional curves that has a point whose removal splits the object into 4 pieces. (c) (4 marks) Consider the following diagram of the Klein bottle, where we glue opposite edges of the rectangle along the arrows. Suppose we cut along the closed loop given by the vertical dotted line in the middle. How many pieces do we get? Choose a piece and give one of its topological properties.

Answer 1

Solutions distostions with nsidos. 1 The number of sides of a Shape is pole served by Stretcthing or bending a shape with n Sides gesult in a Shape A cylinder has a sides while a M obius hand has a I side. Hence the cylinder omd The M obius bomd ale non-equi valut by distootion. Similarly, the number of edges of a shape is poreserved by distortions. Streetling or Bending a shape with nedge result ina shape with nedges. A cylinder has a 2 edge while a 'M' obius hast edge . Hence the cepindel and The 'Mi oblus Bond are mol- equivalent by distosh'on. which the uppee f wherea 2. That point may be the exact central point of the structure nut bolkmay be attached by lower and 677 ond Right portions of the structure are mounted Removal of this point, it com sput into upieces The - 3 Coin *[0.1 with its top and Topologically the Mobius Strip com be defined as the square bottom sides identify by the (8,0) =(1-4,1) for llegal ega 1 as the daciaglene. gelation

A less used poresentations of the moblus ship is as the topological (1). A torus com be quotient of torus con structed as the square 20,13 *[0,1) with edges idenhed as Copy-Cly) Cory N(114) (glue left to right) and (60) sum CX, D (glue bottom to top) it one then also idenfied Caly) - (852) Then one obtains boundary of the mobius Ship and carrier an orbi fold Smucture. which geomeemicaly corresponds to defeleinion - geodesile C Smight liness. in The Mobius elfeut of me edge back into the ship- Spip

Natationally this written as T/s, torus guotiented by the group action of the symmetric group on two letters paint the Switching co-ordinates) and it com be Through of asme configuration space of two unordered point on the circle Sorme ( The edge correspons to the point somes with the torus corresponding to two Possibles the being the point on the circle. Ordered