In this mini-course, we will study topological manifolds of dimension 3. These objects have been widely studied for over a hundred years, and great advances have been made in them, such as G. Perelman’s proof of the Poincaré conjecture. The area of mathematics that deals with 3-manifolds is called Low-Dimensional Topology; in addition to 3-manifolds, other low-dimensional objects are studied, such as knots, surfaces, and even 4-manifolds.
We will begin by reviewing the 2-dimensional case (the Surface Classification Theorem) and then move to dimension 3, where we will prove several 3-manifold decomposition theorems, such as prime factorization, the JSJ decomposition, and the Lickorish-Wallace Theorem on surgeries.
Prerequisites: Having taken a topology course, especially knowledge of the universal property of quotient topology (Review the quotient topology chapter in [M]; see pdf)
[M] Introduction to Topology Class Notes General Topology Topology, 2nd Edition, James R. Munkres: https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-22.pdf
