MATH 7520: Algebraic Topology Homework

MATH 7520   Algebraic Topology

Fall 2013

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*(MgZ) and H*(UhZ2)   (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 S1Z).
· 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
Back to MATH 7520;   to my homepage.