Work all six problems on these sheets; all six count equally towards your grade
1. Calculate the homology groups of the chain complex of abelian groups
where each 
is infinite cyclic with generator 
and the
homomorphisms are given by 
.
2. In 
with coordinates x,y,z,w let 
be
the 3-form
Find 
, where C is the 3-dimensional boundary
of the set
3. Let 
be the 3-sphere
.
a) Prove that there does not exist a submersion (in this case, a smooth map
of rank 1) from 
to 
.
b) Prove that there does not exist a submersion (in this case, a smooth map
of rank 2) from to 
.
4. The smooth manifold M has a covering 
by contractible
open sets such that each intersection 
is contractible
and such that the triple intersection 
.
a) Prove that the de Rham cohomology 
.
b) Give an example to show that 
need not be 0.
c) State a reasonable generalization of a) involving the
condition
.
5. Let M be an n-dimensional smooth manifold. Prove that the tangent bundle and the co-tangent bundle of M are isomorphic as n-dimensional vector bundles.
6. Consider the function 
and the autonomous
differential equation 
with
initial condition 
. Let M be the complete metric space
of continuous curves 
(the metric is 
)
and let
S be the map defined on
M by
a) Prove that S maps M to itself.
b) Prove that S is a contraction map.
c) Start with 
the constant curve 
.
Calculate 
and 
.