General information
Instructor: Wouter van Limbeek
Office: SEO 511
Email: wouter@uic.edu
Book: John M. Lee's Introduction to Smooth Manifolds. Available for free download from the UIC Library here.
Other recommended sources (in no particular order):
1. Guillemin, Pollack: Differential topology.
2. Milnor: Topology from the Differentiable Viewpoint.
3. Hirsch: Differential Topology.
Grades: Blackboard.
Syllabus: Download.
Lectures: MWF, 900-950 AM in AH 303.
Office hours: M, 4-5 pm and T 2-3 pm.
Exams
Midterm: 3/4, in class. Material up to Transversality (p. 143).
Final: 5/5, 1030 am - 1230 pm, AH 303.
Material of the final : Chapters 1-17 of the textbook, except the following sections:
Categories and Functors (in Ch. 3), Group Actions and Equivariant Maps (in Ch. 7),
Time-Dependent Vector Fields and First-Order PDEs (in Ch. 9), Riemannian Metrics (entire Ch. 13), Riemannian Volume Form (in Ch. 15),
Manifolds with Corners, Integration on Riemannian Manifolds, Densities (in Ch. 16), Proof of the Mayer-Vietoris Theorem (in Ch. 17).
Assignments
Due : Beginning of class on date listed on Gradescope.Note: Deadlines are hard deadlines due to posting of solutions.
Numbers refer to end-of-chapter problems in the textbook, NOT mid-chapter exercises.
Homework 1 (due 1/21, solutions): 1-1, 1-3 (you may use Prop. A.16), 1-6, 1-7, 1-9, 1-11.
Homework 2 (due 1/28, solutions): 2-2 (see also Exs 1.8 and 1.34), 2-3, 3-1, 3-6, 3-8 (see p. 72 for definition of velocities and 3.23 for surjectivity).
Homework 3 (due 2/4, solutions): 2-7, 2-10, 2-14, 3-2, 3-3, 3-4.
Homework 4 (due 2/11, solutions): 4-2, 4-4, 4-5(a) (you only need to prove submersivity), 4-6, 4-9, 4-13.
Homework 5 (due 2/18, solutions): 5-1, 5-5, 5-6, 5-10, 5-23, 6-1.
Homework 6 (due 2/25, solutions): 6-2, 6-7 (only the part about 6.26), 6-10, 6-13(a), 6-16(d).
Homework 7 (due 3/11, solutions): 6-5, 6-9 (only the first question) 7-2, 7-4 (instead of B.3, you can also use that diagonalizable matrices are dense; the choice is yours), 8-1, 8-4, 8-16, 8-18.
Homework 8 (due 3/18, solutions): Read Ch. 12 up to `Tensors and Tensor Fields on Manifolds' (p. 316). In addition: 8-25 (Lie(G) is the Lie algebra of G; a Lie algebra is abelian if its bracket is trivial), 9-1(a), 9-6, 9-7, 9-16, 9-17.
Homework 9 (due 4/1), solutions): 10-6, 10-10, 10-15, 10-17, 11-5, 11-7(a), 11-14.
Homework 10 (due 4/8, solutions)): 11-15 (in (c), assume U = Rn), 11-17, 14-1, 14-5, 14-6 (for (b), only do the computation in spherical coordinates), 14-7(a) (use the computation from 11-7(a)), 14-9.
Homework 11 (due 4/15, solutions): 15-2, 15-3 (do not use 15-34), 15-4, 15-5 (only for TM), 15-10, 15-13 (you can use the statement of 15-12), 16-3(a) (for covering maps).
Homework 12 (due 4/22, solutions): 16-2, 16-4, 16-5 (assume M,N are closed), 16-9 (in (a), only show it is an orientation form on the sphere), 17-1, 17-4. Read Section `The Mayer-Vietoris Theorem' in Chapter 17.
Homework 13 (due 4/29, solutions): 17-8, 17-10, 17-12, 17-13.