Solutions 9

1. The answer depends on how you write your letters. For a sans serif font you get the letters grouped something like:
   AR B CGIJLMNSUVWZ DO EFTY H KX PQ
   Thinking of the letters as 2-dimensional one gets B on its own, ADOPQ together and all the rest together.
   The three dimensional classification is the same.

2. Thinking of the two holed torus as a sphere with two "handles" attached, slide one of the handle's ends under the arch formed by the other.
   
   
   Thinking of the two holed torus as a torus with one extra "handle" attached, turn the handle "over the top" as shown.
   
   
   

3. Go round the curves starting with a and when you get back to the vertex take the next available path "going round anticlockwise" -- that is, "turn left" when you get back to the vertex. This will give you the boundary of a 12-gon with edge word
   a b c d^-1 c^-1 d e f^-1 e^-1 f a^-1 b^-1

4. 1. Not a planar model: too many as
      
   2. Not a planar model of a compact surface: c is a free edge. (It is K # T with a disc removed!)
      
   3. Is a planar model of an orientable compact surface
      
   4. Is a planar model of an non-orientable compact surface (b^-1 is matched with itself)
      
   5. Is a planar model of an orientable compact surface
      
   6. Not a planar model of a compact surface: a is a free edge
      
   7. Is a planar model of an non-orientable compact surface
      
   
   

5. Suitable edge words are:
   a b a^-1 b^-1 c d c^-1 d    a b a^-1 b c d c^-1 d   a b a^-1 b^-1 c d c^-1 d^-1 e e

6. 1.  a b c d b e a f g d-1 g-1 h c j f e-1 j h-1
      1 2 2 2 2 2 1 2 1 2  2  1  2 2 2 1  2  2  1

      So F = 1, E = 9, V = 2 ⇒ χ = -6 and so this is a join of 8 projective planes

   2.  a1 a2 a3-1 a4-1 a5 a6-1 a6 a5-1 a4 a3 a2-1 a1-1
      1  2  3  4   5   6  7   6   5   4  3  2   1

      So F = 1, E = 6, V = 7 ⇒ χ = 2 and so this is a sphere

   3.  a1 a2 a3 . . . an a1 a2 a3 . . . an
      1  2  3  . . . n 1  2  3  . . . n  1

      So F = 1, E = n, V = n ⇒ χ = 1 and so this is a projective plane

   4.  a1 a2 a3 a4 a1-1 a2-1 a3-1 a4-1
      1  1  1  1  1   1   1   1    1

      So F = 1, E = 4, V = 1 ⇒ χ = -2 and so this is a join of two tori

   5.  a1 a2 a3 a4 a5 a1-1 a2-1 a3-1 a4-1 a5-1
      1  2  1  2  1  2   1   2   1    2    1

      So F = 1, E = 5, V = 2 ⇒ χ = -2 and so this is also a join of two tori
      
   
   

7. Slicing it as shown and putting in the edges labelled e gives a planar model with edge word
        a b e^-1d^-1c^-1d c e a b
   ⇒ (as above) F = 1, E = 5, V = 3 ⇒ χ = -1 and so this is a join of 3 projective planes.

   You can see this another way by observing that splitting the surface along the dotted curve gives a projective plane with a disc removed and (after turning it inside out!) a torus with a disc removed.