A Generalization of Surgery to Cellular Complexes

In the 1990s Robin Forman carefully generalized Morse theory to cellular complexes. In this thesis we are concerned with the connection Morse theory provides in surgery theory. There exists a theorem linking surgery on embedded spheres to cobordisms whose proof relies on smooth Morse theory. Therefore, we examine if a similar theorem exists on cellular complexes whose proof relies on discrete Morse theory. In order to accomplish this we will provide a way to perform surgery on cellular complexes as well as a possibility for defining cellular cobordism.