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  1. Added an example (dR cohomology for spheres) with a very sketchy proof sketch. More examples, worked out in more detail, to follow. Might make a de Rham cohomology page and move the examples there.

    Mark Moon

    diff, v36, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2020

    added statement of the relation to the PL de Rham complex (here)

    diff, v38, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2023


    Explicitly, given a differential kk-form ω\omega, its de Rham differential dωd\omega can be computed as

    dω(v 0,,v k)= i(1) i v iω(v 0,,v i1,v i+1,,v k)+ iUnknown characterj(1) i+jω([v i,v j],v 0,,v i1,v i+1,,v j1,v j+1,,v k),d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i<j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),

    where v iv_i are vector fields on XX, [,][-,-] is the Lie bracket of vector fields, and L L_{-} is the Lie derivative of a smooth function with respect to a vector field.

    diff, v41, current