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Topology : Euler Characteristic

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Example Question #1 : Euler Characteristic

Calculate the Euler Characteristic given the following Betti numbers.

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

Possible Answers:

$\chi=-1$

$\chi=1$

$\chi =0$

$\chi=2$

None of the answers are correct.

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 : Euler Characteristic

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

Possible Answers:

None of the answers are correct.

$\chi=0$

$\chi=2$

$\chi=-1$

$\chi=1$

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,

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

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 #3 : Euler Characteristic

What is the Euler Characteristic for a line.

Possible Answers:

$\chi=-1$

$\chi=1$

A line doesn't have an Euler Characteristic.

$\chi=2$

$\chi=0$

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,

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

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

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

Report an Error

Example Question #4 : Euler Characteristic

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

Possible Answers:

$\chi=2$

$\chi=4$

$\chi=0$

$\chi=3$

$\chi=1$

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.

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

Report an Error

Example Question #5 : Euler Characteristic

What is the Euler Characteristic for a point?

Possible Answers:

$\chi=0$

$\chi=-1$

$\chi=2$

$\chi=1$

A point doesn't have an Euler Characteristic.

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.

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

Report an Error

Example Question #6 : Euler Characteristic

What is the Euler Characteristic for a triangle?

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 can be calculated therefore,

$\\\beta_0=3\\\beta_1=3 \\\beta_2=1$

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

$\\\chi=3-3+1 \\\chi=1$

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Topology : Euler Characteristic

Study concepts, example questions & explanations for Topology

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Example Question #1 : Euler Characteristic

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