Exercises 9

1. Consider the upper-case letters of the alphabet { A, B, C, D, ... , Z } as being made up of (infinitely thin) lines. Classify them up to topological equivalence.
   If we treat them as being made of lines of finite thickness (so that they are two-dimensional sets) how does the classification change?
   If we treat them as being carved out of (say) wood (so that they are three-dimensional sets) does the classification change again?

   Solution to question 1
   
   

2. Show that the two embeddings of the two-holed torus shown on the right can be deformed into one another in R³.
   [Hint: Think of the two-holed torus as being made from a sphere by attaching two "tubes".]
   
   Show how the two views on the right of a ring and a two-holed torus can be deformed into one another in R³.
   
   Solution to question 2

   
   

3. Sketch the planar model obtained by slicing the space model of T # T # T along the curves a, b, c, d, e, f indicated.
   
   
   Solution to question 3

4. Which of the following are the edge words of planar models of compact surfaces ? For those that are, decide whether the surfaces are orientable or not.
   
   1. a b c b^-1 c^-1 a a^-1 d a d^-1
      
   2. a b a^-1 c b d e d^-1 e^-1
      
   3. a ba^-1 c^-1 b^-1 c
      
   4. a b^-1 c^-1 b^-1 a^-1 c^-1
      
   5. a bc d a^-1 b^-1 c^-1 d^-1
      
   6. a b c d c d^-1 b^-1
      
   7. a b c b a c
      
   
   Solution to question 4

5. If T, K and P are a Torus, Klein bottle and Projective plane, draw planar models and write down edge-words for the surfaces:
       T # K    K # K    T # T # P

   Solution to question 5

6. Calculate the Euler Characteristics of the surfaces whose planar models have the following edge-words and hence give their standard forms as joins of either tori or projective planes.
   
   1. a b c d b e a f g d^-1 g^-1 h c j f e^-1 j h^-1
      
   2. a₁ a₂ a₃^-1 a₄^-1 a₅ a₆^-1 a₆ a₅^-1 a₄ a₃ a₂^-1 a₁^-1
      
   3. a₁ a₂ a₃ . . . a_n a₁ a₂ a₃ . . . a_n
      
   4. a₁ a₂ a₃ a₄ a₁^-1 a₂^-1 a₃^-1 a₄^-1
      
   5. a₁ a₂ a₃ a₄ a₅ a₁^-1 a₂^-1 a₃^-1 a₄^-1 a₅^-1
      
   
   Solution to question 6

   
   

7. A model of a surface is made from the annulus shown (the area between the two squares) by identifying the edges as indicated.
   Show that by adding an extra edge connecting the outer and inner sqares, this can be made into the usual kind of planar model and identify the surface.
   If the surface is sliced along the dotted curve, identify the two pieces that this gives.

   Solution to question 7