Quiz 04

Quiz 04 - (4 The boundary of the disk D 2:= x ∈ R 2 ||| x...

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MATH 74 QUIZ 4 Pick one of the following questions and make a 10-15 minute appointment to present it to me before Wednes- day May 13th. You will also be responsible for any definitions or theorems you use in your proof or expla- nation, though I will only ask for a statement of theorems and perhaps a brief description. (1) Prove that π 1 ( S 1 , x 0 ) = Z . (2) Prove the fundamental theorem of algebra. (You get to assume π 1 ( S 1 , x 0 ) = Z for this one.) (3) Prove that π 1 ( S 2 , x 0 ) = { e } (including a description of stereographic projection) and that π 1 ( V, v 0 ) = { e } for any vector space V . Then describe the fundamental groups of the torus and Klein bottle using group presentations.
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Unformatted text preview: (4) The boundary of the disk, D 2 := { x ∈ R 2 ||| x || < 1 } , is a circle, as is the boundary of the Mobius band. What surfaces do we get when we glue the disk to itself, the disk to a Mobius band, and a Mobius band to itself along these circle boundaries? Are there distinct ways to glue these pieces together? Give an argument using pictures discussed in class. Also give an argument for why such gluings actually yield a surface (starting from the deﬁnition of surface given in class). (5) Prove Kan Kampen’s theorem. (See Hatcher for the statement of the theorem and its proof.) 1...
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