MATH 7590: Cohomology Theory Homework
MATH 7590 Cohomology Theory
Spring 2003
Homework Problems
- 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:S^{2n-1} -> S^{n} and g:S^{n} -> S^{n}, 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:S^{2n-1} -> S^{2n-1} and f:S^{2n-1} -> S^{n}, 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:H_{q}(C (tensor) Z/m) ->
H_{q-1}(C (tensor) Z/m)
be the
Bockstein associated to the exact sequence
0 -> Z/m -> Z/m^{2} -> 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 H_{n}(X,A) are finitely
generated abelian groups for each n. Show that:
- H^{1}(X,A) is a torsion-free abelian group.
- ^{ }H^{n}(X,A) is also finitely generated
- ^{ }rank H^{n}(X,A) = rank H_{n}(X,A)
- ^{ }the torsion subgroup of H^{n}(X,A) is
isomorphic to the torsion subgroup of H_{n-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 RP^{2} x RP^{2} with
Z coefficients, and with Z/2 coefficients (using cellular
homology; redo this problem later using the
Künneth formula).
- If X is a finite CW complex, determine the relationship between
H_{*}(X) and H_{*}(S^{k} x X).
- Problem #2 on Munkres, page 366.
- Determine the cohomology ring of
RP^{2} x RP^{2} with Z/2 coefficients.
- Problems #4, 5, 6 on Munkres, page 393.
- Distinguish CP^{2} and S^{2} v S^{4}
by showing that the two spaces have non-isomorphic cohomology rings.
- Problem #5, parts (b) - (e) on Munkres, page 407.
- For an orientable surface M_{g} (the connected sum of g tori),
find the matrix of the bilinear
form H^{1}(M_{g};R) x
H^{1}(M_{g};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.