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MAT 531 Topology-Geometry II
Midterm Examination

April 11, 1996


tex2html_wrap_inline34 1. This problem is about the tangent bundle to the 2-sphere tex2html_wrap_inline36 . As usual, we write tex2html_wrap_inline38 , where U is the complement of the South pole and V is the complement of the North pole. Show how the tangent bundle admits trivializations over U and V (i.e. isomorphisms to the product bundles tex2html_wrap_inline48 and tex2html_wrap_inline50 ) and exhibit the transition function relating those trivializations over tex2html_wrap_inline52 .

tex2html_wrap_inline34 2. The real projective plane tex2html_wrap_inline56 can be considered as tex2html_wrap_inline58 , where as usual tex2html_wrap_inline60 , and the relation tex2html_wrap_inline62 is tex2html_wrap_inline64 . The map tex2html_wrap_inline66 defined by


defines a map tex2html_wrap_inline70 which is invariant under tex2html_wrap_inline62 , and therefore defines a map tex2html_wrap_inline74 .

a. Show that v is an immersion, i.e. that v has rank 2 at every point of tex2html_wrap_inline56 .

b. Show that v is one-one.

c. Suppose v is composed with projection onto tex2html_wrap_inline86 by suppressing one of the first three coordinates. Is the resulting map still an immersion? Is it one-one?

tex2html_wrap_inline34 3. Let tex2html_wrap_inline90 be a smooth vectorfield defined in an open set tex2html_wrap_inline92 , and suppose S is a smooth, oriented piece of surface lying in U, with boundary C (C inherits an orientation from S as usual). The ``Traditional Stokes' Theorem'' states that the integral over S of the curl of X is equal to the integral of X along C. [Recall that the integral of a vectorfield X over a piece of surface parametrized by tex2html_wrap_inline114 , tex2html_wrap_inline116 is given by tex2html_wrap_inline118 , and that the integral of X along a curve parametrized by tex2html_wrap_inline122 is tex2html_wrap_inline124 .] Show that this theorem is equivalent to a special case of our ``Stokes' Theorem.''


Tony Phillips
Wed Apr 10 19:37:51 EDT 1996