Unformatted text preview: x, y ) = ( ∞ X 1  x ny 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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 Spring '05
 SMITH
 Math, Topology, homotopy equivalent, explicit homotopy equivalence, finite dimensional sphere

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