MATH 7520: Algebraic Topology Homework

## MATH 7520   Algebraic Topology

### Homework Problems

 · 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
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