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# hw11 - Math W4051 Problem Set 11 due Wednesday December 5...

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Math W4051: Problem Set 11 due Wednesday, December 5 Reading: Munkres, Ch.13. Also take a look at Theorem 54.6 in Ch.9. 1. Express the following abelian groups in standard form: (a) Ab x, y | x 2 = y 3 ; (b) Ab x, y | x 2 = y 4 . 2. Let X and Y be manifolds (possibly of different dimensions). Pick x 0 X and y 0 Y. We define their wedge X Y to be the space obtained from the disjoint union of X and Y after identifying x 0 with y 0 : X Y = X Y / x 0 y 0 . Show that π ( X Y, x 0 ) = π 1 ( X, x 0 ) * π 1 ( Y, y 0 ) . 3. Let S 1 sit inside R 3 as { ( x, y, 0) R 3 | x 2 + y 2 = 1 } . Denote by X the complement of S 1 in R 3 . Compute π 1 ( X, 0) . (Hint: Draw a deformation retraction from X to S 2 S 1 . ) 4. We denote by B n the standard open unit ball in R n and by B B n the smaller open ball of radius = 1 / 2 . We also let S = B be the sphere of radius . (a) Let X be a connected manifold of dimension n 3 . Let h : B n U be a homeomor- phism from B n to an open set U X. Show that π 1 ( X - h ( B )) = π 1 ( X ) . (b) Let X 1 , X 2 be connected manifolds of the same dimension n 3 . Let h i : B n U i be a homeomorphism from the unit ball to an open subset U i X i , for i = 1 , 2 . Recall that the space X = ( X 1 - h 1 ( B )) ( X 2 - h 2 ( B )) / h 1 ( x ) h 2 ( x ) for all x S is called a connected sum of
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