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205C_midterm_1

# 205C_midterm_1 - Math 205C Topology Midterm 1 Erin Pearse a...

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Math 205C - Topology Midterm Erin Pearse 1. a) State the definition of an n -dimensional topological (differentiable) manifold. An n -dimensional topological manifold is a topological space that is Haus- dorff, has a countable basis at every point, and is locally Euclidean . That is, every point has a neighbourhood which is homeomorphic to an open set of R n . An n -dimensional differentiable manifold M is an n-dimensional topolog- ical manifold with a differentiable structure . That is, there is a collection of coordinate charts f = { ( U i , ϕ i ) } which cover M such that i) x M, U x f s.t. x U x and U x = V R n ii) For any two charts ( U, ϕ ) and ( V, ψ ) , U V 6 = = ψ ϕ - 1 and ϕ ψ - 1 are diffeomorphisms of ϕ ( U V ) and ψ ( U V ) in R n . iii) U is maximal in the sense that any chart ( U, ϕ ) which compatible with f in the sense of (ii) is included in f . b) Describe all 1-dimensional manifolds. If a 1-dimensional manifold is compact, it is homeomorphic to S 1 . If a 1- dimensional manifold is not compact, it is homeomorphic to R 1 . Anything not falling into either category can readily be shown to be (i) not 1-dimensional, or (ii) not a topological manifold. c) Describe all closed orientable and non-orientable 2-dimensional manifolds. If a 2-dimensional closed manifold is orientable, then it is a sphere, a torus, or a connected sum of tori. That is, it is an n -genus torus (with a sphere corresponding to genus 0). If a 2-dimensional closed manifold is non-orientable, then it is a Klein bottle, projective plane, or connected sum of them. d) What are the fundamental groups of the manifolds in (b) and (c)? π 1 ( S 1 ) = Z and S 1 has universal covering space R 1 . π 1 ( R 1 ) = { 1 } and R 1 is its own universal covering space. π 1 ( S 2 ) = { 1 } and S 2 has universal covering space R 2 . π 1 ( T 2 ) = Z × Z and T 2 has universal covering space R 2 . π 1 (# g i =1 T 2 ) = Z 2 g and # g i =1 T 2 has universal covering space R 2 . π 1 ( K ) = F ( a,b ) { aba - 1 b } and K has universal covering space R 2 . π 1 ( R P 2 ) = Z 2 and R P 2 has universal covering space S 2 .

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2 Math 205C - Topology Midterm Erin Pearse 2. Explain why each of the following figures either is or is not a topological manifold. a) This object is clearly 1-dimensional. Take x to be an inter- section point with open nbd U as shown. Assuming this space has the natural (subspace) topology, U would have to be homeomorphic to an open set V R 1 . Since U - { x } consists of three connected components, and for V - { f ( x ) } con- sists of only two, f : U V cannot be a homeomorphism. This is explained in a little better detail in (v), below. ¥ b) By the same argument as above, any open nbd of the in- tersection point x cannot be homemorphic to an open set of R 1 . ¥ c) Though not a differentiable manifold, this space is clearly homeomorphic to ( a, b ) R 1 and is thus a 1-dimensional topological manifold. ¥ d) This space is not Hausdorff, as any open set containing x also contains y . Since it is not Hausdorff, it fails to be a topological manifold. ¥ e) Any point of this space has a nbd homeomorphic to an open
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205C_midterm_1 - Math 205C Topology Midterm 1 Erin Pearse a...

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