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Topology : Introduction

Study concepts, example questions & explanations for Topology

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Example Questions

Example Question #1 : Betti Numbers

What is the simplex in the zero dimension?

Possible Answers:

Face

Edge

Vertex

Line

Object

Correct answer:

Vertex

Explanation:

In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the zero dimension is known as a vertex.

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Example Question #2 : Betti Numbers

What is the simplex in the second dimension?

Possible Answers:

Vertex

Space

Edge

Line

Face

Correct answer:

Face

Explanation:

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the second dimension is known as a face. What is the simplex in the zero dimension?

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Example Question #3 : Betti Numbers

What is the simplex in the first dimension?

Possible Answers:

Face

Object

Edge

Point

Vertex

Correct answer:

Edge

Explanation:

These are also referred to as:

Zero simplex refers to a point or vertex.

One simplex refers to a line or edge (which is connected by two points)

Two simplex refers to a face (which is connected by three lines)

Three simplex refers to the 3D object created by two faces.

Therefore the simplex in the first dimension is known as a edge.

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Example Question #4 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=30 \\\beta_1=32 \\\beta_2=1$

Possible Answers:

$\chi=-2$

$\chi=3$

$\chi=1$

$\chi=0$

$\chi=-1$

Correct answer:

$\chi=-1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\beta_0=10 \\\beta_1=12 \\\beta_2=20$

$\\\chi=30-32+1 \\\chi=-2+1 \\\chi=-1$

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Example Question #1 : Topology

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=7 \\\beta_1=12\\\beta_2=6$

Possible Answers:

$\chi=1$

$\chi=2$

$\chi=-2$

$\chi=0$

$\chi=-1$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\beta_0=7 \\\beta_1=12\\\beta_2=6$

$\\\chi=7-12+6 \\\chi=-5+6 \\\chi=1$

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Example Question #1 : Introduction

Calculate the Euler Characteristic given the following Betti numbers.

$\\\beta_0=4 \\\beta_1=5 \\\beta_2=2$

Possible Answers:

$\chi =0$

None of the answers are correct.

$\chi=-1$

$\chi=1$

$\chi=2$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\beta_0=4 \\\beta_1=5 \\\beta_2=2$

$\\\chi=4-5+2 \\\chi=-1+2 \\\chi=1$

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Example Question #2 : Introduction

What is the Euler Characteristic associated with a 2-dimensional square?

Possible Answers:

$\chi=-1$

$\chi=2$

$\chi=1$

None of the answers are correct.

$\chi=0$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers can be calculated therefore,

$\\\beta_0=4 \\\beta_1=4 \\\beta_2=1$

now substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\chi=4-4+1 \\\chi=1$

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Example Question #1 : Introduction

What is the Euler Characteristic for a line.

Possible Answers:

$\chi=1$

$\chi=-1$

A line doesn't have an Euler Characteristic.

$\chi=0$

$\chi=2$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers can be calculated therefore,

$\\\beta_0=2 \\\beta_1=1 \\\beta_2=0$

now substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\chi=2-1+0 \\\chi=1$

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Example Question #2 : Introduction

What is the Euler Characteristic associated with a 3-dimensional cube?

Possible Answers:

$\chi=2$

$\chi=0$

$\chi=1$

$\chi=3$

$\chi=4$

Correct answer:

$\chi=2$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers can be calculated therefore,

$\\\beta_0=8 \\\beta_1=12 \\\beta_2=6$

now substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\chi=8-12+6 \\\chi=2$

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Example Question #4 : Introduction

What is the Euler Characteristic for a point?

Possible Answers:

$\chi=0$

A point doesn't have an Euler Characteristic.

$\chi=2$

$\chi=-1$

$\chi=1$

Correct answer:

$\chi=1$

Explanation:

The Euler Characteristic is calculated by the following equation.

$\text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}$

This is also written as,

$\chi=v-e+f$

Recall that the Betti numbers represent the vertices, edges, and faces of the object.

$\\\beta_0=v \\\beta_1=e \\\beta_2=f$

In this particular question the Betti numbers can be calculated therefore,

$\\\beta_0=1 \\\beta_1=0 \\\beta_2=0$

now substitute them into the formula to calculate the Euler Characteristic and solve.

$\\\chi=1-0+0 \\\chi=1$

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Topology : Introduction

Study concepts, example questions & explanations for Topology

All Topology Resources

Example Questions

Example Question #1 : Betti Numbers

Example Question #2 : Betti Numbers

Example Question #3 : Betti Numbers

Example Question #4 : Betti Numbers

Example Question #1 : Topology

Example Question #1 : Introduction

Example Question #2 : Introduction

Example Question #1 : Introduction

Example Question #2 : Introduction

Example Question #4 : Introduction

All Topology Resources

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