Homework 1Due: Friday, October 16, 2009Problem 1Homeomorphisms. (10 points.) Give explicit homeomorphisms to show that the following spaces withtopologies inherited from the respective containing Euclidean spaces are homeomorphic:• R, the real line;• (0, 1), the open interval;• S1− {(0, 1)}, the circle with one point removed.Problem 2Classifying 2-manifolds. (20 points.) Characterize the two surfaces below in terms of genus, boundary,and orientability.1Problem 3Klein bottle. (20 points.) Cut and paste the standard polygonal schema for the Klein bottle (a, a, b, b) toobtain the polygonal schema in which opposite edges of a square are identified (a, b, a−1, b).aabbababProblem 4Order complex. (20 points.) A flag in a simplicial complex K in Rdis a nested sequence of proper faces,σ0< σ1< . . . < σk. The collection of flags form an abstract simplicial complex A sometimes referred to asthe order complex of K. Prove that A has a geometric realization in Rd.Problem 5Alpha complexes. (10 points.) Let S ⊆ Rdbe a finite set of points in general position. Recall thatˇCech(r) and Alpha(r) are theˇCech and alpha complexes for radius r ≥ 0,ˇCech(r) = Nrv{Bx(r)}x∈S, andAlpha(r) = Nrv{Bx(r) ∩ Vorx}x∈S. Is it true that Alpha(r) =ˇCech(r) ∩ Delaunay? If yes, prove thefollowing two subcomplex relations. If no, give examples to show which subcomplex relations are not valid.1. Alpha(r) ⊆ˇCech(r) ∩ Delaunay2.ˇCech(r) ∩ Delaunay ⊆ Alpha(r)Extra creditDeciding Isomorphism. (30 points.) What is the computational complexity of recognizing isomorphicabstract simplicial
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