Course: Math 518, Differentiable Manifolds, Section B1
Time: MWF 9:00--9:50am
Location: 447 Atlgeld Hall
Instructor: Chris Leininger
Phone: 265-6763
Email: clein (at) math.uiuc.edu
Office: 265 Altgeld Hall
Office hours: by apppointment
Text: Differential Topology by V. Guillemin and A. Pollack.
Foundations of Differentiable Manifolds and Lie Groups by F. Warner.
Homework: This will be assigned periodically throughout the semester and usually collected within one or two weeks. Everyone should turn in their own solutions, but I also expect you to work together as well as independently.
An optimal strategy is to: (1) try each problem yourself first; (2) get together with others to discuss your solutions and questions; (3) write up the solutions yourself. There is a balance which must be achieved to optimize the benefits of homework: Struggling with a problem leads to a much deeper understanding of the concepts. On the other hand, if you are really stuck, talking to other can help you get you moving again.
I may give more problems than you can reasonably do, and some may be difficult. Please be sure you can do the more routine problems before tackling the harder ones. Turn in only polished solutions---don't hand in your scratch work!
Tests: There will be both a midterm 10/8 in the evening in 143 AH. You can come anytime from 5 until 9, but I think that 2 hours will easily suffice. There will also be a final exam on 12/15, 1:30--4:30.
Grading: The grades for the course will be determined by homework 20%, midterm 30%, and the final 50%. I will not assign any letter grades until the end of the semester.
Syllabus:Definitions and properties of differentiable manifolds and maps, (co)tangent bundles, vector fields and flows, Frobenius theorem, differential forms, exterior derivatives, integration and Stokes' theorem, DeRham cohomology, inverse function theorem, Sard's theorem, transversality and intersection theory.

The grader has graciously provided solutions to some of the homework problems. The solutions available are linked following the due dates.

Problem set 1. Due 9/3.
Problem set 2. Due 9/15. The notes on group actions (below) may be helpful for working problems 4--7 from this problem set. Note that these last four problems do not need to be turned in.
Problem set 3. Due 9/22.Solutions
Problem set 4. Due 9/29.Solutions
Problem set 5. Complete by 10/6 (this will not be collected).Solutions.
Problem set 6. Read GP Section 2.3. Do problems: Sect. 2.1: 8, 10 ( df_z(n_z) < 0 ), 11; Sect. 2.3: 7 (X should not contain 0), 8. Assume where necessary that your manifolds are embedded in R^n. This will be due on Wednesday (10/15).Solutions.
Problem set 7, due 10/20. Read GP Section 2.2--2.4. Do problems Sect. 2.2: 3, 4; Sect. 2.3: 5, 12, 13, 15, 16, 20 (see 19).Solutions
Problem set 8. Due 10/27.Solutions.
Problem set 9. Due 11/3.Solutions.
Problem set 10, due 11/10. Read: GP Section 3.4, 3.5; Warner Section on Vector Fields, pages 34--41. Do GP problems Sect. 3.4: 5, 8--11. Sect. 3.5: 1 (fix the definition of h_t), 3. Additional problem (1) Suppose f:M --> N is a degree d covering map or compact oriented boundaryless manifolds. Prove that the euler characteristic of M is d times that of N. If f is just assumed to be a degree d map (not necessarily a covering map) is this still true? Explain.Solutions.
Problem set 11. Due 11/19. (note the change in due date.) Solutions
Problem set 12. Due 12/10.Solutions
Problem set 13. Not due. Read: GP Section 4.4--4.8. Do problems: Sect. 4.4: 2, 3, 12; Sect. 4.7: 2,3,4.

Here is a very vague overview of the material we have covered this semester.

Notes on point set topology
Notes on group actions



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