Euler Characteristic - Topology

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Question

Calculate the Euler Characteristic given the following Betti numbers.

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

Answer

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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Question

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

Answer

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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Question

What is the Euler Characteristic for a line.

Answer

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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Question

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

Answer

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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Question

What is the Euler Characteristic for a point?

Answer

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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Question

What is the Euler Characteristic for a triangle?

Answer

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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