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

6. (a) (4 marks) Explain why a cylinder is not equivalent by distortion to a Möbius band. (b) (3 marks) Draw an object made o

Aus- a) The Number of sides of a shape is preserved by distortions stretching or bending a shape With n sides Result in a shape With sides . A Cylindea has a sides While M obius hand has a lside. Hence the Cylinded and the im obius band are Nona Eauivalent by distortion. Similarly, The Number of Edges of a shape is Poreseaved by disto ations. Staretching ou Bending a shape With n Edge white

Result in a shape with n Edges. A Cylinder has 2 Edge While a'm' obius hos ledge Hence the Cylinder and the mi obius band are Not Equivalent by distortion.

of the b) That point may be the exact central point structure Where, a Null bolt May be Attached by which the upper & Lower and Left and right Structure mounted Poortions of the are by the Removal of this point, it can split into upicus. A mobius can be c) Topologically the storp defined as the sauare 6, 1] x [el] With its Top and bottom side s identity by the Relation (aco)~ (~x, 1) For Cleane eaul as the diagrams. A less used presentation of the mobiles Moblus strip is as the Topological aruotient of the tosius Clig A torus can be Constructed

as the sanane 6,13 x6,13 klith Edges identified Cosy) NC14) (glue left to right) as (614) - (1.4) and (xo) - Sin (X, 1) (glue bottom to top) itone then also identifed (nly) - (4,8). Then one obtains boundary of the mobius strip and Caurien an orbitold structure which geometrically Corresponds to Reflection geodesic (straight lines). In the mobius storip back into the Effect off the Edge strip Nationally this written as T2/S2 toous geotenied by the group action of the symmetric group on letters (Switching coordinates) and it can be For Through configuration of Two anordered point on the Two of as the Space Circle possibles the same (The Edge corresponds to the Point being the same) with the Corresponding to toous Two ordered point on the Circle.