Topology : Betti Numbers

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:

Vertex

Object

Line

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 : Introduction

What is the simplex in the second dimension?

Possible Answers:

Space

Edge

Face

Vertex

Line

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 : Introduction

What is the simplex in the first dimension?

Possible Answers:

Object

Face

Edge

Point

Vertex

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 : Introduction

Calculate the Euler Characteristic given the following Betti numbers.

Possible Answers:

Correct answer:

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

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

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

Example Question #5 : Introduction

Calculate the Euler Characteristic given the following Betti numbers.

Possible Answers:

Correct answer:

Explanation:

The Euler Characteristic is calculated by the following equation.

This is also written as,

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

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

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