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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?

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    for starters this is not a simplical decomposition, the verticies w,v are identified so all your triangles collapse. See Munkries Algebraic topology for a simplical decomposition. – 2017-01-20
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    it is a delta complex, and simplicial homology / cohomology can be computed from such a structure. the vertices w and v are not identified in the above complex – 2017-01-20
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    (see Hatcher's Algebraic Topology) – 2017-01-20
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    I would call this cellular cohomology then, rather than simplicial. – 2017-01-20
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    whats a delta complex ? And please dont tell me to look it up in hatcher(which book I dont really like). – 2017-01-20
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    Hi Dan, I follow Hatcher for the course I'm taking, and in it cellular cohomology relates to cell complexes and simplicial relates to delta complexes ? Is this not the norm? – 2017-01-20
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    @Rene Schipperus "Delta-complex" is a term coined by Hatcher himself, so one can find it only in his textbook. However, it's not Hatcher's invention: it's essentially the same thing as a semi-simplicial set, as introduced by Eilenberg and Zilber in 1950. (A semi-simplicial set is a presheaf $\Delta_+^\mathrm{op}\to\mathcal{Set}$, where $\Delta_+$ is the subcategory of $\Delta$ with morphisms being injective maps.) Hatcher's "Delta-complex" is the geometric realization of a semi-simplicial set. – 2017-01-21
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    @Rene Schipperus This particular pedagogical idea of Hatcher is rather nice, as it's easier to build certain things as geometric realizations of (semi-)simplicial sets. – 2017-01-21
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    @DonAlejo thank you for explaining, I was unaware delta complexes were not a universally known thing – 2017-01-21
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    @ReneSchipperus Why don't you like the Hatcher textbook? – 2018-11-10

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