# Math 6520: Differentiable manifolds

## Prof. Allen Knutson TTh 11:40 in Malott 207

The final exam is now posted. It is due on Tuesday the 17th of December. You may use your notes and homeworks, and Guillemin and Pollack, but not other sources (e.g. books, people, internet).

Book: Guillemin and Pollack, supplemented with Bott and Tu.
My office hours in 515 Malott after class on Tuesdays.
TA Pak-Hin Li's office hours in 218 Malott on Wednesdays 10 AM - noon.

HW #1 due Thursday 9/19:

• [GP] Ch 1.1 #9, 17. Ch 1.2 #5. Ch 1.3 #2. Ch 1.4 #2 is false -- fix it and prove that. #10.
• HW #2 due Thursday 9/26:
• [GP] Ch 1.5 #6, 9. Ch 1.6 #3 (trickier than it looks). Ch 1.7 #15. Ch 1.8 #5, 9.
• HW #3 due Thursday 10/3:
• [GP] Ch 2.1 #8, 9. Ch 2.2 #2. Ch 2.3 #10. 2.4 #5,9.
• HW #4 due Thursday 10/10:
• [GP] Ch 2.5 #2,9. 3.2 #3. Do the same for long, finite exact sequences; orienting all but one space puts an orientation on the remaining one. 3.2 #16. 3.3 #10. 3.4 #3.
• HW #5 due Thursday 10/17:
• [GP] Ch 3.3 #4, 11. 3.5 #4, 5, 6. 3.6 #1, 2.
• HW #6 due Thursday Hallowe'en:
• [GP] Ch 4.2 #1. Ch 4.3, the exercise. Ch 4.4 #3,9,10.
• HW #7 due Thursday 11/7:
• Write T2 as a union of two cylinders U,V, whose intersection is itself two cylinders. Use Mayer-Vietoris to compute H*(T2).
• [BT] Exercise 1.7 (p19).
• Use M-V to compute H* of T2 minus two points.
• Use M-V, the previous, and induction, to compute H* of a genus g surface (a doughnut with g holes) minus two points.
• HW #8 due Thursday 11/21.
• Consider the torus T2 as a square modulo identification of the boundary edges, as we've done so often. Let f rotate this square 90 degrees. Compute the action of f* on cohomology, i.e. pick bases of cohomology and write down matrices. (I recommend using the PoincarÃ© duals of some oriented submanifolds.) From there compute the Lefschetz number as the supertrace of the action on cohomology. Confirm that it agrees with the fixed point formula.
• Do the same for the z |-> zn map on CP1.
• Consider the "transpose" action on the square, t(x,y) := (y,x). This induces an action on T2, hence t* on its cohomology. Compute those matrices in the basis from the first problem. Compute the Lefschetz number at least one way. Bearing in mind that L(f) = Euler characteristic if f is homotopic to the identity, show that (1) L(t) = Euler characteristic of T2 but (2) this t is not homotopic to the identity.
• Use Mayer-Vietoris to compute the compactly supported cohomology of the open MÃ¶bius strip.