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We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed. [ABSTRACT FROM AUTHOR]

A theory of (co)homologies related to set-theoretic n-simplex relations is constructed in analogy with the known quandle and Yang-Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalise Hietarinta's idea of "permutation-type" solutions to the quantum (or "tensor") n-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented. [ABSTRACT FROM AUTHOR]

We present a general setting to investigate 풰fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some 풰fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. R. Parthasarathy and K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be 풰fin-cocycle superrigid. [ABSTRACT FROM PUBLISHER]