I am trying to compute directly the simplicial cohomology with Z coefficients of the real projective plane, with the delta complex structure shown below. I have that the 1 - cocyles are maps $\Phi$: < a,b,c > --> Z such that $\Phi$(c) = 0 and $\Phi$(a) = $\Phi$(b) , whereas 1 - coboundaries are $\Phi$ such that $\Phi$(c) = 0 and $\Phi$(a) = $\Phi$(b) = $\Theta$(w) - $\Theta$(v) for some $\Theta$ : < v,w> --> Z.
But then surely every 1 - cocycle is a 1 - boundary so that the 1st cohomology group is trivial? (Which is false) Could somebody explain where the problem is?
