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hw10 - x y = ∞ X 1 | x n-y n | 2 1 2 Let 0 =(0 and let S...

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Math W4051: Problem Set 10 due Wednesday, November 28 Reading: Munkres, Ch.9 § 59-60, Ch.11. 1. Show that x, y | xyxy = 1 is isomorphic to Z * ( Z / 2 Z ) . 2. Show that x, y, z | xyz = zyx is isomorphic to Z * ( Z × Z ) . 3. Show that x, y | x 2 = y 3 is isomorphic to x, y | xyx = yxy . 4. Let n be a positive integer. Show that x, y | x n = 1 , y 2 = 1 , yxyx = 1 is a finite group. How many elements does it have? 5. Munkres, Ch.11 § 73, exercise 1 on p. 445. 6. Show that the complement of a finite set of points in R n is simply connected if n 3 . 7. Let X R 3 be the union of n distinct lines through the origin. Compute π 1 ( R 3 - X ) . Challenge 1 (extra credit): Show that π 1 ( R 2 - Q 2 ) is uncountable. Challenge 2 (extra credit): Recall from the first homework that l 2 denotes the set of sequences x = { x n } of real numbers for which 1 ( x n ) 2 is finite. We defined a metric on l 2 by d
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Unformatted text preview: x, y ) = ( ∞ X 1 | x n-y n | 2 ) 1 / 2 . Let 0 = (0 , , . . . ) and let S ∞ be the unit sphere in l 2 : S ∞ = { x ∈ l 2 | d ( x, 0) = 1 } . Show that S ∞ is homotopy equivalent to a point, by writing down an explicit homotopy equivalence. Remark: A space X is called contractible if it is homotopy equivalent to a point or, equivalently, if the identity id X : X → X is homotopic to a constant map. Clearly a contractible space is simply connected. The converse is false: for n > 1, the finite dimensional sphere S n ⊂ R n +1 is simply connected but not contractible, as it can be shown using more advanced techniques from algebraic topology....
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