# Homology of torus and Klein's bottle

• ddo
In summary: Thank you for your help!In summary, the mathematician is trying to find the singular homology groups of a torus and Klein's bottle. He calculates the zero'th homology group and the first homology group by using the Mayer-Vietoris sequence. He then derives the second homology group by using the left and right maps in the sequence. He argues that the second homology group is \mathbb{Z} because the left map maps the kernel of the right map to 0 and the right map maps the kernel of the middle map to \mathbb{Z}.
ddo

## Homework Statement

I'm trying to calculate singular homology groups of the torus and Klein's bottle using the Mayer-Vietoris sequence.

## The Attempt at a Solution

I represent both spaces as a rectangle with identified edges. Then I take the sets:
U=rectangle without the boundary
V=rectangle without the middlepoint
so U is contractible thus H_n(U)=0 for n>0, H_0(U)=Z
V=S1vS1 so H_1(V)=ZxZ, H_n(V)=0
and their intersection = S1, H_n(S1)=0, H_1(S1)=Z, H_0(S1)=Z
Now from the M-V sequence for n>2 we get an exact sequence
0->0x0->H_n(T)->0, so H_n(T)=0.
But I don't know what to do for smaller n...

Hi ddo!

Do you need to derive everything from the Mayer-Vietoris sequence? That looks quite hard.

The zero'th homology group is very easy since it is the number of path connected components.
The first homology group is also easy by calculating the fundamental group of $S^1\times S^2$...

The second homology group can be derived from Mayer Vietoris then.

I suppose the Mayer-Vietoris hint was there to make the task easier :)
So H_0 is Z because there is only one connected component, H_1 is the abelianization of the fundamental group, both torus and Klein's bottle have abelian fundamental groups so for torus it's $Z \times Z$, for Klein's bottle $Z \times Z_2$.
Now M-V gives:
$0 \to H_2(T) \to Z \to 0 \times (Z \times Z) \to H_1(T) \to Z$
And I still don't know how to derive H_2(T)...

Write out the full Mayer-Vietoris. I'll do it for the torus

$$0\rightarrow H_2(X)\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z} \rightarrow 0$$

Now, it is easy to see that $H_2(X)$ is either 0 or $\mathbb{Z}$.

Now, reason from left to right as follows:
The end of the sequence is

$$\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z} \rightarrow 0$$

So the the map is a surjection, and the kernel of the map is isomorphic to $\mathbb{Z}$.

Consider the next part of the sequence:

$$\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}$$

The kernel of the right map equals the image of the left map. Thus the image of the left map is isomorphic to $\mathbb{Z}$. Thus the kernel of the left map is necessarily 0.

Now consider the next part of the sequence, etc.

Try to complete this argument.

$$\mathbb{Z}\times\mathbb{Z} \rightarrow\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}$$
The kernel of the right map is 0, so the image of the left map is 0, so the kernel of the left map is $\mathbb{Z}\times\mathbb{Z}$
Next:
$$\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}$$
The kernel of the right map is $\mathbb{Z}\times\mathbb{Z}$, so the image of the left map is $\mathbb{Z}\times\mathbb{Z}$, so the kernel of the left map is 0.
$$H_2(X)\rightarrow \mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}$$
The kernel of the right map is 0, so the kernel of the middle map is $\mathbb{Z}$, so the image of the left map is also $\mathbb{Z}$, so $H_2(X)$ can't be 0.
Is that correct?

Sounds good! So the second homology group is $\mathbb{Z}$!

The thing with the Klein Bottle should be analogous!

## 1. What is a torus and a Klein's bottle?

A torus is a mathematical shape that resembles a doughnut, with a hole in the middle. A Klein's bottle is a non-orientable surface, meaning it cannot be distinguished from its mirror image.

## 2. How are torus and Klein's bottle homologous?

Torus and Klein's bottle are homologous because they have the same genus, or number of holes, which is one. This means that they can be continuously deformed into each other without tearing or cutting.

## 3. What is the significance of homology between torus and Klein's bottle?

The homology between torus and Klein's bottle is significant in topology, as it helps us understand the properties and relationships of different shapes and spaces. It also plays a role in various fields of science, such as physics and biology.

## 4. Can a torus and a Klein's bottle be visualized in 3-dimensional space?

Yes, a torus and a Klein's bottle can both be visualized in 3-dimensional space. However, a Klein's bottle may require more complex techniques, such as embedding it in 4-dimensional space, to be fully visualized.

## 5. Are there any real-world examples of torus and Klein's bottle?

Yes, there are real-world examples of torus and Klein's bottle. Examples of torus include a life preserver, a bagel, and a swimming pool ring. A Klein's bottle can be seen in the shape of a Möbius strip, a twisted loop of paper with only one side and one edge.

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