MATH 7590: Cohomology Theory Homework
MATH 7590 Cohomology Theory
- State and prove a Mayer-Vietoris theorem for singular cohomology.
- Compute the cohomology of the n-sphere using induction and
(i) Mayer-Vietoris; (ii) the long exact sequence of a pair.
- Problems #1, 2, 3, 5 on Munkres, page 298.
- 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.
- Let C be a chain complex of free abelian groups, and let
B:Hq(C (tensor) Z/m) ->
Hq-1(C (tensor) Z/m)
Bockstein associated to the exact sequence
0 -> Z/m -> Z/m2 -> Z/m -> 0.
Show that B o B = 0.
- 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:
- H1(X,A) is a torsion-free abelian group.
- 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)
- 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.
- Compute the homology of RP2 x RP2 with
Z coefficients, and with Z/2 coefficients (using cellular
homology; redo this problem later using the
- If X is a finite CW complex, determine the relationship between
H*(X) and H*(Sk x X).
- Problem #2 on Munkres, page 366.
- Determine the cohomology ring of
RP2 x RP2 with Z/2 coefficients.
- Problems #4, 5, 6 on Munkres, page 393.
- Distinguish CP2 and S2 v S4
by showing that the two spaces have non-isomorphic cohomology rings.
- Problem #5, parts (b) - (e) on Munkres, page 407.
- 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.
- Problem #6 on Munkres, page 407.
- Let M be a compact, triangulable, manifold with boundary. Show that the
boundary of M is not a retract of M.