MATH 7590 Cohomology Theory
- 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.
- 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.
- Compute the cohomology of the n-sphere two ways:
- using induction and the long exact sequence of the pair
- using induction and the Mayer-Vietoris sequence in cohomology
- 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
- 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:
- Hn(X,A) is also finitely generated
- rank Hn(X,A) = rank Hn(X,A)
- the torsion subgroup of Hn(X,A) is isomorphic to the torsion subgroup of Hn-1(X,A)
- 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.
- Determine the homomorphism induced by f in integral cohomology, and in mod 2 cohomology.
- Show that there are at least two homotopy classes of maps from
P2 to S2.
- 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.
- 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.
- 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.
- Show that the two definitions of the Hopf invariant given in class are equivalent.
- 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.
- 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.
- Distinguish CP2 and S2 v S4 by showing that the two spaces have non-isomorphic cohomology rings.
- Show that, for n > 0, a homotopy equivalence CP2n --> CP2n must preserve orientation.
- 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.
- 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
Hq(X;Z) and f*:Hq(X;Z) -->
Hq(X;Z) are isomorphisms for all q.
- 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.
- 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.
- 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.
- Show that the intersection of A and B is equal to M.
- Show that Hq(M) is isomorphic to the direct sum of i*(Hq(A)) and j*(Hq(B)) for 0 < q < n-1.
- Show that Hq(A) = Hq(B) = 0 for q greater than or equal to n-1.
- 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.
- 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.
- Let W be a compact manifold with boundary. Show that the boundary of W is not a retract of W.
- Distinguish CP2 # CP2 and S2 x S2 by showing that the two spaces have non-isomorphic cohomology rings.
- 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.
- 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.
- 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.
- Compute the homology of RP2xRP2 with Z coefficients, and with Z/2 coefficients (using the Kunneth formula, and using cellular homology).