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Material Type: Assignment; Class: Geometry and Topology.; Subject: MATHEMATICAL SCIENCE; University: Ball State University; Term: Fall 2009;
Typology: Assignments
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Due: September 18, 2009 Dr. Fischer
(i) Is X a manifold? If yes, of what dimension? (ii) Is X connected? (iii) Is X compact?
(i) The vertices of the plane model end up being identified to how many distinct points of X? (ii) Is X a manifold? If yes, is it orientable or non-orientable?
T 1 = {v 1 , v 2 , v 4 }, T 2 = {v 1 , v 3 , v 5 }, T 3 = {v 2 , v 3 , v 5 }, T 4 = {v 2 , v 4 , v 5 }, T 5 = {v 3 , v 4 , v 6 }, T 6 = {v 1 , v 2 , v 6 }, T 7 = {v 4 , v 5 , v 6 }, T 8 = {v 1 , v 5 , v 6 }, T 9 = {v 1 , v 3 , v 4 }, T 10 = {v 2 , v 3 , v 6 }.
Assemble these triangles into a plane model for X. Can you tell what surface it is?
[See the reverse side for hints!]
Hints:
ParametricPlot[{Sin[t], Sin[2t]}, {t, 0, 2Pi}]
(i) If you remove the interior of a disk from P 2 , then the remainder is topo- logically equivalent to a M¨obius band. To see this, think of P 2 as being obtained from the standard unit disk in the Euclidean plane by identifying every pair of diametrically opposite points to one point. Show that under these identifications, the set {(x, y) ∈ R^2 | y ≤ − 1 /2 or y ≥ 1 / 2 } becomes topologically equivalent to a disk. It is the interior of that disk that you want to remove. (ii) Figure 2.6 on page 34 of the text shows a dark shaded subset of K^2 which is a copy of the M¨obius band. Show that all of the light shaded part taken together is a copy of the M¨obius band as well. (iii) Conclude that each of P 2 #P 2 and K^2 is the union of two M¨obius bands glued together along their (one) boundary circle. Hence the two spaces are topologically equivalent, since they can be obtained in the same way.