The definition of a manifold is fairly unrestrictive — we just need the area around each point to look like (be homeomorphic to) Euclidean space. However, with that definition we can say a lot about what manifolds are possible. In two dimensions, we can use the polygonal representation discussed in class along with the “zipping” technique to completely classify compact surfaces that do not have a boundary into three different groups. In three dimensions, manifolds can be split into different pieces that each have one of eight geometries. Higher dimensions have interesting results as well, such as something called the surgery technique for manifolds of dimension 5 and higher. Interestingly, manifolds of dimension 4 (which is also the dimension of space-time) are the hardest to classify. I find it very interesting that, given the local definition of a manifold, we can make such global assertions about its structure. I am not sure which dimension/classification I want to focus on yet, and welcome any suggestions.