Instructor: Ljudmila Kamenova

Office: Math Tower 3-115

Office hours: Tue-Thu 2:00-3:00pm in Math Tower 3-115; drop by my office anytime or send me an e-mail: kamenova@math.stonybrook.edu.

Grader: Matthew Huynh

Grader's office hours: Matthew Huynh's card

Course Description:

Foundations of differentiable manifolds: differentiable maps, vector fields and flows, and differential forms and integration on manifolds. Stokes' theorem. Froebenius theorem. Lie derivatives. Immersions and submersions. DeRham chomology, cochain complexes, degree of a map, Mayer-Vietoris Theorem. Fiber bundles, Ehresmann's theorem. Higher homotopy groups and the homotopy exact sequence. Lifting of paths.

Textbooks:

• Introduction to Smooth Manifolds (2nd Ed.), by John M. Lee, Springer 2012.

• Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2000.

• See Brightspace for additional links and lecture notes.

Grading: Homework accounts for 50% of the total grade; the Midterm is worth 20% of the total grade; the Final is worth 30% of the total grade.

• The midterm exam will be held on Thursday, March 26th, in class. It will include the material from Chapters 8, 9, 10, 11, 12 and 14 in Lee's book.

• The final exam will be held on Thursday, May 14th, from 11:15am to 1:45pm in the classroom Physics P130.

Week.	Lecture Dates.	Topics (tentative).
1.	Jan 27th and 29th.	Vector fields, Lie brackets.
2.	Feb 3rd and 5th.	Integral curves and flows. Lie derivatives.
3.	Feb 10th and 12th.	Vector bundles, subbundles, sections. The cotangent bundle.
4.	Feb 17th and 19th.	Fiber bundles, Ehresmann's theorem.
5.	Feb 24th and Feb 26nd.	Higher homotopy groups. The notion of fibration. The homotopy exact sequence of a fibration, applications.
6.	March 3rd and 5th.	Tensors, differential forms.
7.	March 10th and 12th.	Distributions and Foliations. The Frobenius Theorem.
Spring Break (March 16 - 22).
8.	March 24th and 26nd.	Review and Midterm.
9.	March 31st and April 2nd.	Orientations revisited, integration on manifolds.
10.	April 7th and 9th.	De Rham cohomology and computations. Degree of a map, revisited. Mayer-Vietoris Theorem.
11.	April 14th and 16th.	Singular and cellular co/homology, relation to de Rham (statement only), calculations and applications.
12.	April 21st and 23rd.	Poincare duality (Mark McLean). Obstructions, part I (guest lecture on Thursday - Dennis Sullivan).
13.	April 28th and 30th.	Poincare duality - applications (Mark McLean). Obstructions, part II (guest lecture on Thursday - Dennis Sullivan).
14.	May 5th and 7th.	Review, catch-up, extra topics if time permits.

See Matthew Huynh's Brightspace announcement with instructions on how to upload your homework via Gradescope. Homework is a fundamental part of this course, and you will have to work hard on the assigned problems in order to succeed. Late homework will not be accepted. Homework will account for 50% of the total grade. The exercises will be from the course textbook. Starred questions are for extra credit, and therefore optional.

Number.	Due Date.	Exercises.
1.	Tuesday, Feb 10th, via Gradescope.	[Lee's book] 8-3, 8-4, 8-11, 8-13, 8-16.
2.	Tuesday, Feb 17th, via Gradescope.	[Lee's book] 9-3, 9-4, 9-5, 9-8*, 9-18*.
3.	Tuesday, March 3rd, via Gradescope.	[Lee's book] 10-6, 10-7, 11-7, 11-9, 11-10*.
4.	Tuesday, March 10th, via Gradescope.	[Lee's book] 12-8, 14-1, 14-3*, 14-5, 14-6.
5.	Tuesday, April 14th, via Gradescope.	[Lee's book] 15-1, 15-3, 15-5, 15-8*.
6.	Tuesday, April 21st, via Gradescope.	[Lee's book] 16-3 (a), 16-4 (you may assume the retraction is smooth here), 16-5, 16-6*.

Accessibility Support Center (SASC) Statement: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact the Student Accessibility Support Center (SASC), ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

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