In this last week we will discuss presentations of groups, and how the CW complex structure gives rise to descriptions of the fundamental groups of surfaces. We will see that for every group there is a surface whose fundamental group is isomorphic to that group, and will conclude with the classification of surfaces. In the final lecture we will briefly mention the Universal Cover and give a revision.

Learning outcomes

After this week, you should be able to

  • describe groups by giving generators and relators
  • compute the fundamental group of higher-genus surfaces
  • explain how to derive the fundamental group of a CW complex
  • explain how to derive a surface whose fundamental group coincides with a given group

Tasks and Materials

  • Have a look at the lecture notes when they appear.
  • Read pages 63 – 70 in Hatcher’s book.
  • Make sure you understood most of the content of the lecture, and let me know if something is unclear