| · | Use the Universal Coefficient Theorem to determine the cohomology groups of RPn with coefficients in Z, and with coefficients in Z2. |
| · | Hatcher, page 205, problems 6, 8, and 11 (with m=2 and n=1) |
| · | For Mg=T#...#T and Uh=RP2#...#RP2, determine the ring structure of H*(Mg; Z) and H*(Uh; Z2) (see Hatcher, page 228, problem 1) |
| · | Compute the homology and cohomology of RP2 x RP2 with coefficients in Z, and with coefficients in Z2. |
| · | Hatcher, page 229, problem 11 |
| · | Determine the structure of the cohomology ring H*(S1 x S1 x S1; Z). |
| · | Show that the two definitions of the Hopf invariant given in class are equivalent. |
| · | If f is a map from S2n-1 to Sn and g is a map from Sn to Sn, show that the Hopf invariant of the composite g ° f is equal to the square of the degree of g times the Hopf invariant of f. |
| · | If h is a map from S2n-1 to S2n-1 and f is a map from S2n-1 to Sn, show that the Hopf invariant of the composite f ° h is equal to the degree of h times the Hopf invariant of f. |
| · | Suppose p:E ---> M is a covering space, where M and E are connected n-manifolds. If M is orientable (that is, Z-orientable), show that E is orientable, and that every covering transformation preserves orientation. |
| · | For Uh=RP2#...#RP2, identify the orientable 2-fold covering space. |
| · | Hatcher, page 257, problems 2, 5, 6, 7, 8, 9 |
| · | Hatcher, page 259, problems 16, 17, 24, 25 |
| · | Hatcher, page 260, problems 28, 29, 32, 33, 34 |
| · | Let T 3 = S 1 x S 1 x S 1 and f : T 3 ---> T 3. Find conditions which ensure that f has a fixed point. How about f : T n ---> T n? |
| · | For X and Y path connected, show that πn(X x Y) is isomorphic to πn(X) x πn(Y). |
| · | Hatcher, page 358, problems 1, 2, 12 |
Dan Cohen Fall 2013