The Prime Decomposition Theorem for 3-Manifolds

A 3-manifold is a space which locally looks like R^3. A major theme in 3-manifold Topology is to understand and classify 3-manifolds. Given two compact 3-manifolds M_1,M_2 we can form another 3-manifold by taking what’s called the “connect sum” of M_1 and M_2. Under this operation, 3-manifolds can be decomposed uniquely into prime pieces just like the integers can be decomposed uniquely as a product of primes. We will discuss this prime decomposition theorem for 3-manifolds while also giving a wide variety of examples.