Infinite-order combinatorial Transverse Intersection Algebra TIA via the probabilistic wiggling model
Daniel An, Ruth Lawrence, Dennis Sullivan
TL;DR
This work constructs an infinite-dimensional, differential graded algebra (a dga over $\mathbb{Q}$) of transverse intersections on cubical complexes by a probabilistic wiggling of cell boundaries. Starting from 1D, via a wiggling model, it derives a concrete algebra generated by wiggled points ${\emptyset}_a^{m,n}$ and wiggled intervals $x_{a,b}^{m,n}$ with explicit boundary maps and intersection rules, then tensors this theory across dimensions to obtain a $d$-dimensional TIA as a product of 1D theories. The main contributions include a fully explicit 1D model with decorated generators and precise intersection coefficients, an algebraic framework that restores the product rule for the boundary, and connections to Whitney forms and fluid-algebra discretizations, offering a robust discrete analogue of differential forms with infinite-order lattice corrections. Practically, this provides a rigorous combinatorial toolkit for modeling transverse intersections and could enhance stable computations in fluid dynamics and discrete geometric analysis.
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.