Homology group computations with kunneth formula and universal coefficient theorem

[antiemptyv]

Homework Statement

I am to compute the homology groups $$H_*(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2)$$, with coefficients in $$\mathbb{Z}_2$$.

Homework Equations

Kunneth formula and universal Coefficient theorem

The Attempt at a Solution

First I need the homology groups, obtained via the Kunneth formula:
$$H_0(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}$$,
$$H_1(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}_2$$,
$$H_2(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z} \oplus \mathbb{Z}$$,
$$H_3(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}_2$$,
$$H_4(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z} \oplus \mathbb{Z}$$, and
$$H_i(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong 0$$ for $$i > 4$$.

Now the Universal Coefficient Theorem yields the following:
$$H_0(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2) \cong \mathbb{Z}_2$$,
$$H_1(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2) \cong \mathbb{Z}_2$$,
$$H_2(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2) \cong \mathbb{Z} \oplus \mathbb{Z}$$,
$$H_3(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2) \cong \mathbb{Z}_2$$,
$$H_4(\mathbb{C}P^2 \times \mathbb{R}P^2; \mathbb{Z}_2) \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$$, and
$$H_i(\mathbb{C}P^2 \times \mathbb{R}P^2); \mathbb{Z}_2 \cong 0$$ for $$i > 4$$.

I'm not sure if these are correct, and H_4 seems weird, having three summands.

 

[mighty2000]

I would appreciate any feedback on my solution. Thank you!

Dear student,

Your solution is almost correct, but there is a slight mistake in your computation of H_2(\mathbb{C}P^2 \times \mathbb{R}P^2). The correct result is H_2(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2, since the Kunneth formula for the second homology group involves the tensor product of H_1(\mathbb{C}P^2) and H_1(\mathbb{R}P^2), both of which are isomorphic to \mathbb{Z}_2. This also affects the computation of H_4(\mathbb{C}P^2 \times \mathbb{R}P^2), which should be \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2.

Other than that, your solution is correct and your use of the Kunneth formula and Universal Coefficient Theorem is appropriate. Keep up the good work!