Topology : Introduction

Study concepts, example questions & explanations for Topology

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Betti Numbers

What is the simplex in the zero dimension?

Possible Answers:

Vertex

Line

Object

Face

Edge

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.

Example Question #2 : Betti Numbers

What is the simplex in the second dimension?

Possible Answers:

Vertex

Space

Line

Edge

Face

Correct answer:

Face

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

Example Question #3 : Betti Numbers

What is the simplex in the first dimension?

Possible Answers:

Object

Point

Edge

Vertex

Face

Correct answer:

Edge

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 first dimension is known as a edge. 

Example Question #4 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

\(\displaystyle \\\beta_0=30 \\\beta_1=32 \\\beta_2=1\)

Possible Answers:

\(\displaystyle \chi=-2\)

\(\displaystyle \chi=-1\)

\(\displaystyle \chi=3\)

\(\displaystyle \chi=0\)

\(\displaystyle \chi=1\)

Correct answer:

\(\displaystyle \chi=-1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=10 \\\beta_1=12 \\\beta_2=20\)

\(\displaystyle \\\chi=30-32+1 \\\chi=-2+1 \\\chi=-1\)

Example Question #5 : Betti Numbers

Calculate the Euler Characteristic given the following Betti numbers.

\(\displaystyle \\\beta_0=7 \\\beta_1=12\\\beta_2=6\)

Possible Answers:

\(\displaystyle \chi=0\)

\(\displaystyle \chi=-2\)

\(\displaystyle \chi=1\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi=-1\)

Correct answer:

\(\displaystyle \chi=1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=7 \\\beta_1=12\\\beta_2=6\)

\(\displaystyle \\\chi=7-12+6 \\\chi=-5+6 \\\chi=1\)

Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

\(\displaystyle \\\beta_0=4 \\\beta_1=5 \\\beta_2=2\)

Possible Answers:

\(\displaystyle \chi=-1\)

\(\displaystyle \chi=1\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi =0\)

None of the answers are correct.

Correct answer:

\(\displaystyle \chi=1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=4 \\\beta_1=5 \\\beta_2=2\)

\(\displaystyle \\\chi=4-5+2 \\\chi=-1+2 \\\chi=1\)

 

Example Question #2 : Euler Characteristic

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

Possible Answers:

\(\displaystyle \chi=-1\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi=0\)

None of the answers are correct.

\(\displaystyle \chi=1\)

Correct answer:

\(\displaystyle \chi=1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=4 \\\beta_1=4 \\\beta_2=1\)

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

\(\displaystyle \\\chi=4-4+1 \\\chi=1\)

Example Question #3 : Euler Characteristic

What is the Euler Characteristic for a line.

Possible Answers:

\(\displaystyle \chi=0\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi=-1\)

\(\displaystyle \chi=1\)

A line doesn't have an Euler Characteristic. 

Correct answer:

\(\displaystyle \chi=1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=2 \\\beta_1=1 \\\beta_2=0\)

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

\(\displaystyle \\\chi=2-1+0 \\\chi=1\)

Example Question #4 : Euler Characteristic

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

Possible Answers:

\(\displaystyle \chi=4\)

\(\displaystyle \chi=1\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi=0\)

\(\displaystyle \chi=3\)

Correct answer:

\(\displaystyle \chi=2\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=8 \\\beta_1=12 \\\beta_2=6\)

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

\(\displaystyle \\\chi=8-12+6 \\\chi=2\)

Example Question #5 : Euler Characteristic

What is the Euler Characteristic for a point?

Possible Answers:

\(\displaystyle \chi=-1\)

A point doesn't have an Euler Characteristic.

\(\displaystyle \chi=1\)

\(\displaystyle \chi=2\)

\(\displaystyle \chi=0\)

Correct answer:

\(\displaystyle \chi=1\)

Explanation:

The Euler Characteristic is calculated by the following equation.

\(\displaystyle \text{Euler Characteristic}=\text{ vertices}-\text{edges}+\text{faces}\)

This is also written as,

\(\displaystyle \chi=v-e+f\)

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

\(\displaystyle \\\beta_0=v \\\beta_1=e \\\beta_2=f\)

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

\(\displaystyle \\\beta_0=1 \\\beta_1=0 \\\beta_2=0\)

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

\(\displaystyle \\\chi=1-0+0 \\\chi=1\)

Learning Tools by Varsity Tutors