Chapter 6: Some results about 3- and 4-dimensional manifolds

The next step in our study of theta functions is a more detailed analysis of the discrete Fourier transform defined by an element of the mapping class group. We will be able to relate this discrete Fourier transform to 3- and 4-dimensional topology. To enpower the reader with the necessary background, we devote this chapter to reviewing the necessary facts from low-dimensional topology. For more details the reader can consult [Rolfsen (2003)]

All manifolds are orientable, and for an oriented manifold M, -M denotes the manifold with orientation reversed. To distinguish easily between 3- and 4-dimensional manifolds, we use for the latter the boldface script. Sⁿ and Bⁿ denote the n-dimensional sphere and ball respectively.