| · | Use the Universal Coefficient Theorem to determine the cohomology groups of RPn with coefficients in Z, and with coefficients in Z2. |
| · | Use the cellular map f from RP2 to S2 which collapses the 1-cell to a point to show that the splitting in the cohomology Universal Coefficient Theorem is not natural with respect to homomorphisms induced by continuous maps. |
| · | 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). |
| · | Distinguish the spaces S2 x S4 and CP3 |
| · | 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. |
| · | Identify the "orientation" double cover for a general non-orientable surface. |
| · |
Suppose p:E → M is a covering space, with E and M connected n-manifolds. If M is orientable, show that E is orientable, and that every covering transformation preserves orientation. |
| · | Hatcher pages 257 - 259, problems 2, 5 - 9, 16, 17, 24, 25. |
| · |
Let M be a closed, connected, orientable n-manifold, and f : M → M a continuous map.
If f* : Hn(M;Z) → Hn(M;Z) is an isomorphism, show that f* : Hq(M;Z) → Hq(M;Z) and f* : Hq(M;Z) → Hq(M;Z) are isomorphisms for all q. |
| · | Hatcher, page 260, problems 28, 29, 32, 33, 34. |
| · | For X and Y path connected, show that πn(X x Y) is isomorphic to πn(X) x πn(Y). |
Dan Cohen Fall 2015