Need Full Solutions with Explanations - Advanced Geometry and Topology Study Guide

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Good day, I need an expert tutor to provide crystal-clear, 100% correct solutions with detailed explanations to ALL the questions in the attachments. This is for my advanced geometry / topology course. I don't understand anything in this course, not even what the questions are asking. Thus, I need someone to define and explain all the complicated math terms and math symbols used in these questions and provide simple, easy-to-understand explanations on how to solve each question that even a 10-year old child would be able to understand it. I only have a high school junior understanding of math, thus these questions are too difficult to understand for me. Therefore, I need someone to provide 100% correct solutions with 100% clear explanations. I need to be able to understand and study from the explanations.

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a like pair. There is no significance in where the sequence begins. So a torus could be written as aba-16-1 or ba-16-1 a and a Möbius Band as abac, baca etc. For each of the following draw the corresponding polygon with identified edges and express it in the form mT?#nD2 or mP2#nD2 where D2 represents a disk, T2 represents a torus and p2 represents a projective plane. .abca-dedo-1, .abca-ded-16-1 6. The following polygon with identified edges represents a surface S. • Label the vertices so that vertices have the same label if and only if they represent the same point on S. • Find the Euler Characteristic of S. • Is S orientable? Give reasons. • Express S, up to homeomorphism, in the form mD2#nT?#r P2 • Are the values of m, n and r unique? • Carry out a single piece of surgery (one cut and a subsequent joining along one identified pair of edges) to obtain a polygon with identified edges which also represents S but which has an adjacent like pair. 7. Consider a 14-gon. Identify the surface. 1. What is topological invariant? Name two different topological invariants. 2. Compute the Euler characteristic of the cylinder • the Mobius band .K . S2 • 12 • P2 3. Investigate what happens to the Euler characteristic when you take con- nected sums. If x(S1) = a and x(S2) = b, what is x(Su#S2)? Prove your answer. 4. Prove that the two Theorems are equivalent: Theorem A: Any closed surface is homeomorphic either to (a) The sphere (b) A connected sum of tori (c) A connected sum of projective planes Theorem B: Any closed surface is homeomorphic either to (a) The sphere (b) A connected sum of tori (c) A connected sum of a tori and one projective plane (d) A connected sum of a tori and two projective planes 5. A PIE (polygon with identified edges) can be represented as a word in which each symbol, I, occurs at most twice, either as an 2 and an It or as two copies of I. Symbols which occur only once represent unidentified edges and those that occur in a pair represent a pair of identified edges with 1 and 2-1 indicating an unlike pair and a couple of I's indicating 1 1. What is topological invariant? Name two different topological invariants. 2. Compute the Euler characteristic of the cylinder • the Mobius band .K . S2 • 12 • P2 3. Investigate what happens to the Euler characteristic when you take con- nected sums. If x(S1) = a and x(S2) = b, what is x(Su#S2)? Prove your answer. 4. Prove that the two Theorems are equivalent: Theorem A: Any closed surface is homeomorphic either to (a) The sphere (b) A connected sum of tori (c) A connected sum of projective planes Theorem B: Any closed surface is homeomorphic either to (a) The sphere (b) A connected sum of tori (c) A connected sum of a tori and one projective plane (d) A connected sum of a tori and two projective planes 5. A PIE (polygon with identified edges) can be represented as a word in which each symbol, I, occurs at most twice, either as an 2 and an It or as two copies of I. Symbols which occur only once represent unidentified edges and those that occur in a pair represent a pair of identified edges with 1 and 2-1 indicating an unlike pair and a couple of I's indicating 1
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Explanation & Answer

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Problem 1:
Topological invariant can be defined as any property of the topology space which is invariant
under homeomorphisms.

Problem 2:
For the Mobius band, the subdivision has 8 vertices, 16 edges, and 8 faces as shown below:

Hence, the Euler characteristic  of the Mobius band is calculated by

  Mobius band   V  E  F  8 16  8  0
For K 2 , its subdivision has 16 edges, 8 vertices, and 8 faces as shown below:

Hence, the Euler characteristic  of K 2 is calculated by

  K 2   V  E  F  8  16  8  0
For S 2 , the sphere is homeomorphic to the cube

which has 8 vertices, 12 edges and 6 faces. Hence, Euler characteristic of the sphere S 2 is

  V  E  F  8  12  6  2
For T 2 , which is presented by the rectangle with edge identifications, with 1 vertex, 2 edges,
and 1 face. Hence, the Euler characteristic  of T 2 is

  V  E  F  1 2 1  0
For P 2 , we label all vertices, edges, and faces as follow

Since there are 8 vertices, 15 edges, and 8 faces, the Euler characteristic of P 2 is

  V  E  F  8  15  8  1
Problem 3:

We will prove that   S1 # S2   a  b  2 . Consider a triangulation T1 of S1 and T2 of S 2 .Let’s
remove the 2 triangles t1 and t 2 where t1  T1 and t2  T2 and glue along the boundaries of t1 and

t 2 . Then, we get a triangulation of S1 # S2 induced by T1 and T2 .
Let V be the number of vertices in this triangulation, then V  v1  v2  3 where vi is the
number of vertices of Ti for i 1, 2 because 1 vertex of t1 is identified to 1 vertex of t 2 .
Similarly, let E be the number of edges in the triangulation induced by T1 and T2 , then

E  e1  e2  3 where ei is the number of edges of Ti for i 1, 2 because 1 edge of t1 is
identified to 1 edge of t 2 .
Lastly, the number of triangles (or face) in that triangulation is equal to F  f1  f 2  2 , where

f i is the number of triangles (or face) in Ti for i 1, 2 since the 2 triangles t1 and t 2 are
rem...