MATH 7590: Cohomology Theory Homework

## MATH 7590   Cohomology Theory

### Homework Problems

1. State and prove a Mayer-Vietoris theorem for singular cohomology.

2. Compute the cohomology of the n-sphere using induction and
(i) Mayer-Vietoris;   (ii) the long exact sequence of a pair.

3. Problems #1, 2, 3, 5 on Munkres, page 298.

4. Show that the two definitions of the Hopf invariant given in class are equivalent.

5. If f:S2n-1 -> Sn and g:Sn -> Sn, show that the Hopf invariant of the composition g o f is equal to the square of the degree of g times the Hopf invariant of f.

6. If h:S2n-1 -> S2n-1 and f:S2n-1 -> Sn, show that the Hopf invariant of the composition f o h is equal to the degree of h times the Hopf invariant of f.

7. Let C be a chain complex of free abelian groups, and let B:Hq(C (tensor) Z/m) ->
Hq-1(C (tensor) Z/m) be the Bockstein associated to the exact sequence
0 -> Z/m -> Z/m2 -> Z/m -> 0. Show that B o B = 0.

8. Suppose that (X,A) is a pair of topological spaces for which the singular homology groups Hn(X,A) are finitely generated abelian groups for each n. Show that:

1. H1(X,A) is a torsion-free abelian group.
2.  Hn(X,A) is also finitely generated
3.  rank Hn(X,A) = rank Hn(X,A)
4.  the torsion subgroup of Hn(X,A) is isomorphic to the torsion subgroup of Hn-1(X,A)

9. If X and Y are finite CW complexes, show that the Euler characteristic of X x Y is equal to the product of the Euler characteristics of X and Y.

10. Compute the homology of RP2 x RP2 with Z coefficients, and with Z/2 coefficients (using cellular homology; redo this problem later using the Künneth formula).

11. If X is a finite CW complex, determine the relationship between H*(X) and H*(Sk x X).

12. Problem #2 on Munkres, page 366.

13. Determine the cohomology ring of RP2 x RP2 with Z/2 coefficients.

14. Problems #4, 5, 6 on Munkres, page 393.

15. Distinguish CP2 and S2 v S4 by showing that the two spaces have non-isomorphic cohomology rings.

16. Problem #5, parts (b) - (e) on Munkres, page 407.

17. For an orientable surface Mg (the connected sum of g tori), find the matrix of the bilinear form H1(Mg;R) x H1(Mg;R) -> R with respect to an appropriate basis.

18. Problem #6 on Munkres, page 407.

19. Let M be a compact, triangulable, manifold with boundary. Show that the boundary of M is not a retract of M.