# Homology group computations with kunneth formula and universal coefficient theorem

1. Apr 20, 2009

### antiemptyv

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Kunneth formula and universal Coefficient theorem

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Can you offer guidance or do you also need help?
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