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
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 .