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# hw1 - Computational Topology Homework 1(due Fall 2009 1...

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Computational Topology Homework 1 (due 9 / 29 / 09) Fall 2009 1. Recall that a simple closed curve is polygonal if its image is the union of a ﬁnite number of line segments. A polygon is the closure of the interior of a simple closed polygonal curve. The boundary of a polygon P is denoted P . (a) Let G n be a regular n -gon centered at the origin, with one vertex at ( 1,0 ) . Describe a homeomorphism φ n : IR 2 IR 2 such that φ n ( G n ) = S 1 . (b) Describe an algorithm for the following problem: Given a simple polygon P , construct a homeomorphism φ : P G n such that φ ( P ) = G n . The input polygon P is represented as an array of n vertices in (say) counterclockwise order. [Hint: Any polygon with holes can be triangulated in O ( n log n ) time. How do you want to represent the output homeomorphism?] (c) Describe an algorithm to construct a homeomorphism φ : IR 2 IR 2 such that φ ( P ) = G n . Together with part (a), this proves the Jordan-Schönﬂies theorem for polygons. 2. A cycle in a topological space X is a continuous map γ : S 1 X . A (free) homotopy between cycles γ and δ in X is a continuous function h : [ 0,1 ] × S 1 X , such that h ( 0, θ ) = γ ( θ ) and h ( 1, θ ) = δ ( θ ) for all θ S 1 . Two cycles are freely homotopic if there is a free homotopy between them. (a) Loops and cycles are almost identical—any cycle can be turned into a loop by choosing a

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hw1 - Computational Topology Homework 1(due Fall 2009 1...

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