Question

EXERCISE 2.6.6 Which of the following words represent legitimate gluing instructions for a plane model of a surface? And if a surface, is it orientable? a. abac ajaza a3a4d

Answer 1

with edge word aba b-1 For the torus we havea 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 b a Again all four corners of the square get identified to the same point. or . For the Projective Plane we have a with edge word abab or a In this case the corners of the square get identified to a pair of points 4. A cylinder has edge word abac and the Mobius band has edge word abac When we make a 2-holed torus we remove discs from tori with edge words abab1 and cde 1d to get surfaces with holes: aba1b le and cie 1d1/which we then join together to get a model with planar word abalb-1cdc1d1