Infinite-order combinatorial Transverse Intersection Algebra TIA via the probabilistic wiggling model

Abstract

This paper constructs a graded-commutative, associative, differential Transverse Intersection Algebra TIA {on the torus (in any dimension) with its cubical decomposition by using a probabilistic wiggling interpretation. This structure agrees with the combinatorial graded intersection algebra (graded by codimension) defined by transversality on pairs of `cuboidal chains' which are in general position. In order to define an intersection of cuboids which are not necessarily in general position, the boundaries of the cuboids are considered to be `wiggled' by a distance small compared with the lattice parameter, according to a suitable probability distribution and then almost always the wiggled cuboids will be in general position, producing a transverse intersection with new probability distributions on the bounding sides. In order to make a closed theory, each geometric cuboid appears in an infinite number of forms with different probability distributions on the wiggled boundaries. The resulting structure is commutative, associative and satisfies the product rule with respect to the natural boundary operator deduced from the geometric boundary of the wiggled cuboids. This TIA can be viewed as a combinatorial analogue of differential forms in which the continuity of space has been replaced by a lattice with corrections to infinite order. See the comparison to Whitney forms at the end of the paper. For application to fluid algebra we also consider the same construction starting with the $2h$ cubical complex instead of the $h$ cubical complex. The adjoined higher order elements will be identical to those required in the $h$ cubical complex. The $d$-dimensional theory is a tensor product of $d$ copies of the one-dimensional theory.