· | Use the Universal Coefficient Theorem to determine the cohomology groups of RP^{n} with coefficients in Z, and with coefficients in Z_{2}. |
· | Hatcher, page 205, problems 6, 8, and 11 (with m=2 and n=1) |
· | For M_{g}=T#...#T and U_{h}=RP^{2}#...#RP^{2}, determine the ring structure of H^{*}(M_{g}; Z) and H^{*}(U_{h}; Z_{2}) (see Hatcher, page 228, problem 1) |
· | Compute the homology and cohomology of RP^{2} x RP^{2} with coefficients in Z, and with coefficients in Z_{2}. |
· | Hatcher, page 229, problem 11 |
· | Determine the structure of the cohomology ring H^{*}(S^{1} x S^{1} x S^{1}; Z). |
· | Show that the two definitions of the Hopf invariant given in class are equivalent. |
· | If f is a map from S^{2n-1} to S^{n} and g is a map from S^{n} to S^{n}, 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 S^{2n-1} to S^{2n-1} and f is a map from S^{2n-1} to S^{n}, 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 U_{h}=RP^{2}#...#RP^{2}, 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 π_{n}(X x Y) is isomorphic to π_{n}(X) x π_{n}(Y). |
· | Hatcher, page 358, problems 1, 2, 12 |
Dan Cohen Fall 2013