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.

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?

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. 

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.

$\displaystyle \\\beta_0=10 \\\beta_1=12 \\\beta_2=20$

$\displaystyle \\\chi=30-32+1 \\\chi=-2+1 \\\chi=-1$

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.

$\displaystyle \\\beta_0=7 \\\beta_1=12\\\beta_2=6$

$\displaystyle \\\chi=7-12+6 \\\chi=-5+6 \\\chi=1$

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.

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

$\displaystyle \\\chi=4-5+2 \\\chi=-1+2 \\\chi=1$

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.

$\displaystyle \\\chi=4-4+1 \\\chi=1$

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.

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

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.

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

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.

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