Today's exercises

to be turned in at the end of the class (as extra credit)
  1. In the following five exercises determine in how many different ways is each of the following 3-manifolds a bundle and describe each.
    • T3
    M is a 3-manifold obtained from a cube by identifying the cube's sides in the following way:
    • front to back with a side to side flip
      top to bottom with a side to side flip
      left and right normally
    • front to back with a side to side flip
      top to bottom with a front to back flip
      left to right normally
    • top to bottom and left to right normally
      front to back with a side to side flip
    • front to back with a side to side flip
      top to bottom with half a turn
      left to right normally
  2. A solid doughnut is topologically D2 x S1. How would you construct a solid Klein bottle?
  3. Describe two different ways in which S2 x I can be glued to get an S2 bundle over a circle.
  4. Here is a gluing diagram for a surface S.



    First determine what this surface is topologically. Then form S x I and glue the opposite vertical sides to each other normally.
    In how many different ways can you glue the top to the bottom to get an S bundle over a circle.