Hi, everyone: I am reading a small expository paper on properties of CP2, in which the intersection form is defined as an integral of the wedge of two forms $w_1$, $w_2$, and these forms $w_1$, $w_2$ (no problem with compact support, since CP2 is compact) seem to have been obtained from the fundamental class [z] of H2(CP2)-- a copy of CP1 (embedded in CP2), after which we integrate $w:=w1\wedge w2$ to get the intersection number.
I am curious on whether I am reading the above correctly, i.e., that the volume form in CP2 is obtained by using the fund. class [z] in H2. If not, would someone explain; if this is correct, if we are we using some form of deRham's theorem to turn a purely topological object like [z] into an object like $w$, for which we must have a differentiable structure defined)?
Thanks in Advance.