dif8-page10 - is a Lie group, determine its dimension....

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Further Exercises ===================================================================================== Exercise 8-1. The configuration space of the pentagon (closed chain of five rods in the plane) with one edge fixed is a compact surface (sometimes with singularities). What kind of surfaces can we obtain? 0 Exercise 8-2. Give an example of a set X with a C -compatible atlas A on it such that the topology induced on X by A is (i) not Hausdorff; (ii) not second countable. Exercise 8-3. Show that the special unitary group * SU(n) = { A e Gl(n, C ) : A A = I, det A = 1 }
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Unformatted text preview: is a Lie group, determine its dimension. Prove that SU(2) is diffeomorphic to 3 the 3-dimensional sphere S . Exercise 8-4. Which surface shall we get from the classification list if we glue to the sphere k > 1 M o bius bands and l handles? Exercise 8-5. Let P be a complex polynomial of degree k having k different 2 roots. Consider the subset of C defined by 2 l M = {(z,w) e C : z = P(w)}. Show that M is diffeomorphic to a sphere with g handles with N points omitted. Express g and N in terms of k and l. 10...
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This note was uploaded on 01/08/2012 for the course MATH 203 taught by Professor Sedita during the Spring '09 term at SUNY Stony Brook.

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