Topology : Euler Characteristic

Study concepts, example questions & explanations for Topology

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

Possible Answers:

None of the answers are correct.

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 #2 : Euler Characteristic

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

Possible Answers:

None of the answers are correct.

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 can be calculated therefore,

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

Example Question #3 : Euler Characteristic

What is the Euler Characteristic for a line.

Possible Answers:

A line doesn't have an Euler Characteristic. 

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 can be calculated therefore,

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

Example Question #4 : Euler Characteristic

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

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 can be calculated therefore,

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

Example Question #5 : Euler Characteristic

What is the Euler Characteristic for a point?

Possible Answers:

A point doesn't have an Euler Characteristic.

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 can be calculated therefore,

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

Example Question #6 : Euler Characteristic

What is the Euler Characteristic for a triangle?

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 can be calculated therefore,

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

Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: