*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 .

Tue May 7 15:51:04 EDT 1996