Math 651: Topology 1
Advanced topics in topology. Topics may include, but are not limited to, piecewise linear topology, 3-manifold theory, homotopy theory, differential topology, Riemannian geometry, and geometric group theory.
Desired Learning Outcomes
Math 651 and 652 present advanced topics in topology. Topics are picked according to the developing interests of the research community and may also be helpful for students studying algebraic geometry or dynamical systems.Topics may include, but are not limited to, the topology of low-dimensional manifolds, piecewise linear topology, geometric group theory, and homotopy theory. For research preparation, the student should also consider courses in algebraic topology, differential topology, and Riemannian geometry.
Students should learn the foundational theorems in their fields of interest. Students should learn to find and read research papers in those fields. Students should learn to solve challenging problems, develop proofs of theorems on their own, and present those proofs clearly and coherently with appropriate illustrative examples.
Minimal learning outcomes
Desired outcomes vary with the topics to be considered.
Low-dimensional manifolds: The student should understand the classification of 2-dimensional surfaces according to orientation, boundary components, and Euler characteristic and should be able to identify a surface by calculating these invariants. The student should know how to represent a 3-manifold by face-pairing, by Heegaard splitting, and by Dehn surgery, how to pass from one representation to another, and how to manipulate these representations. The student should learn the statements of Dehn’s Lemma, the loop theorem, and the sphere theorem, should learn a proof of at least one of these, and should be able to apply these theorems in a number of settings. The student should be able to apply arguments from homology theory, cohomology theory, fundamental groups, and covering spaces in the setting of 3-manifolds. The student should learn the JSJ decomposition theorem and Haken-Waldhausen theory. The student should learn about the eight Thurston geometries and have examples of manifolds represented by each geometry. The student should learn the statements of the Thurston-Perelman Geometrization Theorem. The student should learn to calculate manifold and knot invariants.
Piecewise linear topology: Students should know the definitions of polyhedron and piecewise-linear map. They should be able to manipulate convex and linear combinations of vectors. They should be able to manipulate simplicial complexes, cell complexes, triangulations, simplicial maps, subdivisions, joins, collars. They should be able to use regular neighborhoods. They should understand how to put a simplicial map in general position and employ cut and paste arguments. They should know how to manipulate handles in handle decompositions of manifolds.
Geometric group theory: The basic connection of group theory with geometry follow from two sources, namely, the Cayley graph of a group and the action of a group on a geometry. The student should understand the definition of Cayley graph and should be able to construct the Cayley graph of many groups. The student should know many examples of group actions on geometric spaces, in particular on Euclidean, spherical, and hyperbolic geometries. The student should learn how to build a CW complex having prescribed fundamental group. The student should understand the fundamental group of a graph and its implications for free non Abelian groups. The student should learn how to build the Cayley graph of a free product, free product with amalgamation, HNN extension. The student should understand the fundamental decision problems in groups (the word problem, the conjugacy problem, and the isomorphism problem), their unsolvability in general, and their solvability generically.
Homotopy theory: The student should learn the definitions of the higher homotopy groups and their relationships to CW complexes, covering spaces, homology, and cohomology. The student should learn about fiber bundles and the associated long exact sequences in homotopy. The student should learn the basic homotopy properties of the fundamental spaces in topology, namely, balls, spheres, projective spaces, Stiefel and Grassmann manifolds, the orthogonal, unitary, and symplectic groups. This is a good place to introduce the student to braid groups and mapping class groups.
Possible textbooks for this course include (but are not limited to):
- For 3-manifold theory, the standard references are Hempel, 3-Manifolds, Princeton University Press; Thurston, Three-Dimensional Geometry and Topology, Volume 1, Princeton University Press; and Scott, The Geometries of 3-Manifolds, Bull. London Math. Soc. 15 (19983), 401-487.
For piecewise-linear topology, the standard reference is Rourke and Sanderson, Piecewise Linear Topology, Springer-Verlag.
For geometric group theory, standard references are Bridson, Martin R., and Haefliger, André: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999; Lyndon, Roger C.; Schupp, Paul E. Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001; and an introduction in J. W. Cannon, Geometric Group Theory, in Handbook of Geometric Topology, Elsevier Science B. V., 261-305.
For homotopy theory, the standard references are Hatcher, Algebraic Topology, Chapter 4, Cambridge University Press, and Whitehead, Elements of homotopy theory, Springer-Verlag.
Courses for which this course is prerequisite
Math 652, Topology 2.