Topology Geometry II Final Examination

Stony Brook, May 7 1996, Noon - 3PM

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 tex2html_wrap_inline45 is infinite cyclic with generator tex2html_wrap_inline47 and the homomorphisms are given by tex2html_wrap_inline49 .

2. In tex2html_wrap_inline51 with coordinates x,y,z,w let tex2html_wrap_inline55 be the 3-form


Find tex2html_wrap_inline59 , where C is the 3-dimensional boundary of the set


3. Let tex2html_wrap_inline65 be the 3-sphere tex2html_wrap_inline67 .

a) Prove that there does not exist a submersion (in this case, a smooth map of rank 1) from tex2html_wrap_inline69 to tex2html_wrap_inline71 .

b) Prove that there does not exist a submersion (in this case, a smooth map of rank 2) from tex2html_wrap_inline69 to tex2html_wrap_inline75 .

4. The smooth manifold M has a covering tex2html_wrap_inline79 by contractible open sets such that each intersection tex2html_wrap_inline81 is contractible and such that the triple intersection tex2html_wrap_inline83 .

a) Prove that the de Rham cohomology tex2html_wrap_inline85 .

b) Give an example to show that tex2html_wrap_inline87 need not be 0.

c) State a reasonable generalization of a) involving the condition
tex2html_wrap_inline89 .

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 tex2html_wrap_inline99 and the autonomous differential equation tex2html_wrap_inline101 with initial condition tex2html_wrap_inline103 . Let M be the complete metric space of continuous curves tex2html_wrap_inline107 (the metric is tex2html_wrap_inline109 ) 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 tex2html_wrap_inline121 the constant curve tex2html_wrap_inline123 . Calculate tex2html_wrap_inline125 and tex2html_wrap_inline127 .

Tony Phillips
Tue May 7 15:51:04 EDT 1996