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Homology group computations with kunneth formula and universal coefficient theorem

  1. Apr 20, 2009 #1
    1. The problem statement, all variables and given/known data

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

    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:
    [tex]H_0(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}[/tex],
    [tex]H_1(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}_2[/tex],
    [tex]H_2(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z} \oplus \mathbb{Z}[/tex],
    [tex]H_3(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z}_2[/tex],
    [tex]H_4(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong \mathbb{Z} \oplus \mathbb{Z}[/tex], and
    [tex]H_i(\mathbb{C}P^2 \times \mathbb{R}P^2) \cong 0[/tex] for [tex]i > 4[/tex].

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

    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
     
  2. jcsd
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