Mayer-Vietoris sequence

**harbottle** · October 30th 2009, 07:56 PM

Hi, I have really appreciated the help understanding homology over the last week or two. I have a couple more questions, 28 and 29 again from 2.2 in Hatcher;

For the first part of 28 need to use the M-V sequence to find the homology groups of the space obtained by attaching the boundary circle of a möbius strip to the torus along the circle $\{x_0\}\times S^1$ . Then the second part is the same question but we glue the boundary of the strip to the subset $\mathbf{R}P^1 \subset \mathbf{R}P^2$

I know basically how the MV sequence works, but I am not confident in using it for these spaces, visualisation of which I am finding daunting, especially the part about the real projective planes.

Q29 is very similar. We need to find the homology groups of the surface obtained by gluing the interiors of two surfaces of genus g (ie the compact spaces bounded by such) along the boundary, Mg. Also we need to find the relative homology groups of (R,Mg) where R is the "interior" of Mg.

Still a bit lost on these so any tips or anything would be appreciated, thanks!

**aliceinwonderland** · October 31st 2009, 02:44 PM

Originally Posted by harbottle

Hi, I have really appreciated the help understanding homology over the last week or two. I have a couple more questions, 28 and 29 again from 2.2 in Hatcher;

For the first part of 28 need to use the M-V sequence to find the homology groups of the space obtained by attaching the boundary circle of a möbius strip to the torus along the circle $\{x_0\}\times S^1$ . Then the second part is the same question but we glue the boundary of the strip to the subset $\mathbf{R}P^1 \subset \mathbf{R}P^2$

I know basically how the MV sequence works, but I am not confident in using it for these spaces, visualisation of which I am finding daunting, especially the part about the real projective planes.

Q29 is very similar. We need to find the homology groups of the surface obtained by gluing the interiors of two surfaces of genus g (ie the compact spaces bounded by such) along the boundary, Mg. Also we need to find the relative homology groups of (R,Mg) where R is the "interior" of Mg.

Still a bit lost on these so any tips or anything would be appreciated, thanks!

This is for Q28.

Let X be our target space.
Let A be a torus with a neighborhood of a Mobius band including an attached intersection. Let B be a mobius band with a neighborhood of a torus including an attached intersection.
Then $X = A \cup B$ ; $A \cap B$ is homotopy equivalent to a circle (Note that this is a degree 2 attaching map). Now we are ready to apply the Mayer-Vietoris sequence.

$\cdots \rightarrow H_2(A) \oplus H_2(B) \rightarrow H_2(X) \rightarrow H_1(A \cap B) \rightarrow$ $H_1(A) \oplus H_1(B) \rightarrow H_1(X) \rightarrow H_0(A \cap B) \rightarrow H_0(A) \oplus H_0(B) \rightarrow \cdots$ .

It follows that

$\cdots \rightarrow \mathbb{Z} \oplus 0 \rightarrow H_2(X) \rightarrow \mathbb{Z} \rightarrow$ $\text{ } (\mathbb{Z} \oplus \mathbb{Z}) \oplus \mathbb{Z} \rightarrow H_1(X) \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \cdots$ .

You need to fill in the details here. Note that A deformation retracts to torus and B deformation retracts to a Mobius band. A mobius band in turn deformation retracts to a circle, so $\pi_1(M)=H_1(M)=\mathbb{Z}$ , where M is a mobius band. You also check surjectivity and injectivity of each arrow to find $H_2(X)$ and $H_1(X)$ . Especially, you need to check the degree 2 map between $\mathbb{Z} \rightarrow ((\mathbb{Z} \oplus \mathbb{Z}) \oplus \mathbb{Z})$ , where the image is generated by (0,1, -2). You can find the similar case in here.

From this exact sequence and using the first isomorphism theorem for groups, we have $H_2(X)=\mathbb{Z}$ , $H_0(X)=\mathbb{Z}$ , and $H_1(X) = \mathbb{Z} \oplus \mathbb{Z}$ , which is a group generated by x, y, and z having relations y-2z=0.
================================================== ==
Part (b) is parallel to part (a).

Let X be our target space.
Let A be a RP^2 with a neighborhood of a Mobius band including an attached intersection. Let B be a mobius band with a neighborhood of a RP^2 including an attached intersection.
Then $X = A \cup B$ ; $A \cap B$ is homotopy equivalent to a circle. The attaching map is a degree 2 map.

$\cdots \rightarrow H_2(A) \oplus H_2(B) \rightarrow H_2(X) \rightarrow H_1(A \cap B) \rightarrow$ $H_1(A) \oplus H_1(B) \rightarrow H_1(X) \rightarrow H_0(A \cap B) \rightarrow H_0(A) \oplus H_0(B) \rightarrow \cdots$ .

It follows that

$\cdots \rightarrow 0 \rightarrow H_2(X) \rightarrow \mathbb{Z} \rightarrow$ $\text{ } ((\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}) \rightarrow H_1(X) \rightarrow \mathbb{Z} \rightarrow \cdots$ .

You also need to fill in the details here. Especially, check injectivity and surjectivity of each arrow and degree 2 map between $\mathbb{Z} \rightarrow ((\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z})$ , where the image is generated by (1, -2). Then we have $H_2(X)=0$ , $H_0(X)=\mathbb{Z}$ , and $H_1(X)=\mathbb{Z}/4\mathbb{Z}$ , which is a group generated by x and y having relations 2x=0 and x-2y=0.
================================================
Note: I don't guarantee that the above is error-free. Compare this to your own solution and let me know if you find any error.

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