Geodesic
Dual Homology for the de Rham Cohomology
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 237-246

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A divergence operator $\operatorname {div}$ acting on the graded space $\Omega _*(M)$ of smooth multivector fields on a smooth manifold $M$ is defined. This operator turns $\Omega _*(M)$ into a chain complex defining the usual homology of $M$. 
@article{TM_2004_247_a16,
     author = {E. G. Sklyarenko},
     title = {Dual {Homology} for the de {Rham} {Cohomology}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {237--246},
     publisher = {mathdoc},
     volume = {247},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_247_a16/}
}
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E. G. Sklyarenko. Dual Homology for the de Rham Cohomology. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 237-246. http://geodesic.mathdoc.fr/item/TM_2004_247_a16/