## MATH 7590   Cohomology Theory

### Homework Problems

1. For the "orientation double cover" p:E --> X, identify the covering space E in the case where X is the connected sum of n projective planes.

2. Let p:E --> X be a covering space, and assume that X and E are connected n-manifolds. If X is orientable, show that E is also orientable, and that every covering transformation is orientation preserving.

3. Compute the cohomology of the n-sphere two ways:
1.  using induction and the long exact sequence of the pair
2.  using induction and the Mayer-Vietoris sequence in cohomology

4. Let (X,A) be a pair, and suppose that A is a retract of X. Show that Hn(X) is isomorphic to the direct sum of Hn(A) and Hn(X,A).

5. In this problem, the coefficient ring is the integers, R=Z.
For any pair (X,A), show that H1(X,A) is a torsion-free abelian group.
If (X,A) is a pair such that Hn(X,A) is a finitely generated abelian group for every n, show that:
1.  Hn(X,A) is also finitely generated
2.  rank Hn(X,A) = rank Hn(X,A)
3.  the torsion subgroup of Hn(X,A) is isomorphic to the torsion subgroup of Hn-1(X,A)
6. Let P2 denote the real projective plane. Recall that P2 may be realized as a cell complex with one 0-cell, one 1-cell, and one 2-cell. Define a map f:P2 --> S2 by collapsing the 1-cell to a point.
1.  Determine the homomorphism induced by f in integral cohomology, and in mod 2 cohomology.
2.  Show that there are at least two homotopy classes of maps from P2 to S2.
3.  Use the map f to show that the splitting in the exact sequence given by the universal coefficient theorem
is not natural with respect to homomorphisms induced by continuous maps.

7. Determine the structure of the cohomology rings H*(Tg;Z) and H*(Uh;Z/2) using representative chains and cochains for the homology and cohomology generators as we did for the torus and projective plane in class.

8. Let f:Tg --> Vg T be the map from the orientable surface of genus g to a bouquet of g tori defined in class. Use the map f to determine the structure of the cohomology ring of Tg.

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

10. 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.

11. 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.

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

13. Show that, for n > 0, a homotopy equivalence CP2n --> CP2n must preserve orientation.

14. Determine the structure of the ring H*(RPn;Z). Suggestion: use the fact that the natural homomorphism Z -->> Z/2 induces a surjective map from the cellular cochain complex of RPn with Z coefficients to the cellular cochain complex of RPn with Z/2 coefficients.

15. Let X be a compact, connected, orientable n-manifold, and f:X --> X a continuous map. If f*:Hn(X;Z) --> Hn(X;Z) is an isomorphism, show that f*:Hq(X;Z) --> Hq(X;Z) and f*:Hq(X;Z) --> Hq(X;Z) are isomorphisms for all q.

16. For an orientable surface Tg, find the matrix of the bilinear form H1(Tg;R) x H1(Tg;R) --> R with respect to an appropriate basis.

17. Let A be a tautly imbedded subspace of R3. If the integral homology of A, H*(A;Z), is finitely generated, show that H*(A;Z) and H*(A;Z) are torsion-free.

18. Let M be a connected, closed (n-1)-dimensional submanifold of the n-sphere Sn. By Alexander duality, M is orientable, and Sn \ M has two (open) components, with closures A and B such that A U B = Sn. Let i:M --> A and j:M --> B denote the inclusions.
1.  Show that the intersection of A and B is equal to M.
2.  Show that Hq(M) is isomorphic to the direct sum of i*(Hq(A)) and j*(Hq(B)) for 0 < q < n-1.
3.  Show that Hq(A) = Hq(B) = 0 for q greater than or equal to n-1.
4.  If F is a field, show that i*(Hq(A;F)) is isomorphic to Hom(j*(Hn-q-1(B),F) for 0 < q < n-1.
5.  Conclude that RPn does not imbed in Sn+1, CPn does not imbed in S2n+1, and HPn does not imbed in S4n+1 for n at least 2.

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

20. Distinguish CP2 # CP2 and S2 x S2 by showing that the two spaces have non-isomorphic cohomology rings.

21. 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.

22. Let B:Hn-1(RPn Z/2) --> Hn(RPn Z/2) be the Bockstein associated to the exact sequence 0 --> Z/2 --> Z/4 --> Z/2 --> 0.
Show that B is an isormorphism if n is even, and that B is zero if n is odd.

23. If K and L are finite cell complexes, show that the Euler characteristic of K x L is equal to the product of the Euler characteristics of K and L.

24. Compute the homology of RP2xRP2 with Z coefficients, and with Z/2 coefficients (using the Kunneth formula, and using cellular homology).