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Math Help - Homology of S^{1} one point union S^{2}

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    Homology of S^{1} one point union S^{2}

    I'm working out the homology groups of the one point union of the circle and the sphere.

    The obvious CW complex is 1 0-cell (the point where the circle and sphere touch), 1 1-cell (the circle) and 1 2-cell (the sphere).

    So we get a sequence
    0 -> Z -> Z -> Z -> 0
    d_2 d_1 d_0
    H_{0}=Z

    The first homology group, I think is Z, because the kernel of d_1 is Z and d_2 has degree 0 and so im d_2 = 0.

    Then the second homology group, ker(d_2)/Z=0 as ker(d_2) + im d_2 = Z, and the map from 0->Z has image Z.

    Is this correct?
    Last edited by robeuler; April 2nd 2009 at 07:44 PM.
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