Step 1: We begin this course with a review of fundamental groups (see also section 1.1 ) and the Deck theorem; along with a carefully presented motivational example . (See the Math 54 syllabus for an account of the ideas from basic topology that you will need.) In this example we will articulate what we will mean by giving a space a geometric structure in the context of Euclidean geometry. Hopefully we will finish this course on the same note but using a many holed torus and hyperbolic geometry!
Step 2: The groups of interest. We will get a feel for the groups that form "deck-like" group actions. In particular we will learn about free products and how to view a group in terms of its presentation. Topologically our primary tool to accomplish this understanding will be Van Kampen's theorem (see section 1.2 ). In the process we will classify all compact connected surfaces, and learn how to construct a topological space with a specified fundamental group via the Cayley complex.
Step 3: Covering Space Theory. Including: general liftings, the Deck theorem's converse, and the Galois correspondence (see also section 1.3 ). We will emphasize carefully constructing covers of the punctured torus and a genus two surface. This material will finish the topological half of the course, here is the first exam .
Step 4: Our next goal will be to develop the language with which we shall articulate hyperbolic geometry. The approach taken here will be to use the language of the complex plane. Namely we will examine conformal structures and mappings, and then we tackle the problem of explicitly understanding the properties of the continuous group of conformal mapping of the Riemann Sphere: the Mobius group. In the process we shall explore carefully its subgroups consisting of maps preserving the unit disk and the upper-half plane.
Step 5. Now we develop the most important geometry for topologist; hyperbolic geometry. We will take Reimann's point of view and develop the Poincare disk and upper half plane models via the use of a Riemannian metric. We will explore carefully the primary objects: geodesics, circles, triangles (finite and beyond), horocirlces, bananas, and divine lengths; as well as understand the congruences: finite rotations, rotations about infinity, translations, glide reflections , and reflections. Here are some example problems .
Step 6: In this final step, we discuss the Killing-Hopf Theorem which assures us that each compact surface admits a geometric structure, and then we articulate the notion of this geometry's uniqueness . In particular we put together the geometry and topology we have learned in order to whiteness geometric structures on all compact connected surfaces and the tessellations of the hyperbolic space associated to most of them.