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Assignment Set 3 | Geometry and Topology | MATHS 441, Assignments of Mathematics

Material Type: Assignment; Class: Geometry and Topology.; Subject: MATHEMATICAL SCIENCE; University: Ball State University; Term: Fall 2009;

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2009/2010

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Exercise Set III MATHS 441
Due: September 18, 2009 Dr. Fischer
1. Consider the three spaces described in Exercise 2.2 on page 30 of the text.
For each of these three spaces Xanswer the following questions:
(i) Is Xa manifold? If yes, of what dimension?
(ii) Is Xconnected?
(iii) Is Xcompact?
2. Do Exercise 2.3 on page 34 of the text.
3. Consider the four plane models of Figure 2.11 on page 40 of the text.
For each of these plane models answer the following questions about the space X,
which they represent:
(i) The vertices of the plane model end up being identified to how many
distinct points of X?
(ii) Is Xa manifold? If yes, is it orientable or non-orientable?
4. The following collection of 10 triangles (each denoted by their three vertices)
represents a triangulation of a compact surface X:
T1={v1, v2, v4}, T2={v1, v3, v5}, T3={v2, v3, v5},
T4={v2, v4, v5}, T5={v3, v4, v6}, T6={v1, v2, v6},
T7={v4, v5, v6}, T8={v1, v5, v6}, T9={v1, v3, v4},
T10 ={v2, v3, v6}.
Assemble these triangles into a plane model for X.
Can you tell what surface it is?
5. Suppose that Yis a compact surface with the property that X#Yis topologi-
cally equivalent to Xfor every compact surface X.
Show that Ymust be topologically equivalent to a 2–dimensional sphere.
6. Prove directly (without using Example 3.8) that the connected sum of two
projective planes is topologically equivalent to a Klein bottle: P2#P2K2.
[See the reverse side for hints!]
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Exercise Set III MATHS 441

Due: September 18, 2009 Dr. Fischer

  1. Consider the three spaces described in Exercise 2.2 on page 30 of the text. For each of these three spaces X answer the following questions:

(i) Is X a manifold? If yes, of what dimension? (ii) Is X connected? (iii) Is X compact?

  1. Do Exercise 2.3 on page 34 of the text.
  2. Consider the four plane models of Figure 2.11 on page 40 of the text. For each of these plane models answer the following questions about the space X, which they represent:

(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?

  1. The following collection of 10 triangles (each denoted by their three vertices) represents a triangulation of a compact surface X:

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?

  1. Suppose that Y is a compact surface with the property that X#Y is topologi- cally equivalent to X for every compact surface X. Show that Y must be topologically equivalent to a 2–dimensional sphere.
  2. Prove directly (without using Example 3.8) that the connected sum of two projective planes is topologically equivalent to a Klein bottle: P 2 #P 2 ≈ K^2.

[See the reverse side for hints!]

Hints:

  1. You might want to use Mathematica to make a sketch of the last of the three spaces. Here is the command:

ParametricPlot[{Sin[t], Sin[2t]}, {t, 0, 2Pi}]

  1. First familiarize yourself with the notation. For example, triangle T 1 has vertices v 1 , v 2 and v 4 and meets triangle T 3 only in the vertex v 2 , whereas triangle T 1 and triangle T 4 share a common edge between their vertices v 2 and v 4. Now begin by “laying down” any one of these triangles in the Euclidean plane. Then, one by one, always take a triangle that can be matched up along a free edge with what you have already assembled in the plane so far. If two edges of triangles that have already been laid out are to be identified, label and direct them accordingly.
  2. This problem has a very short solution.
  3. Argue in three steps:

(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.