By matching edges of a point-cloud complex to minimising geodesics and minimizing lifted distances, one can label each cycle by its ambient homology class.
Nerves, fibers and homotopy groups.Journal of Combinatorial Theory, Series A, 102(1):88–93, 2003
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Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings
By matching edges of a point-cloud complex to minimising geodesics and minimizing lifted distances, one can label each cycle by its ambient homology class.