The combinatorial transverse intersection algebra

Abstract

This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by transversality on the usual chains which are in general position. This, (with extra elements added) on the entire $h$-cubulated three torus whose differential satisfies the product rule and which agrees with the set theoretic intersection product appropriately weighted. The construction is characterized given these properties (see Comprehensive Theorem below). The challenge is to minimally adjoin infinitesimal elements when the geometric elements have glancing but transversal intersections weighted in such a way that the associativity (and commutativity) is not destroyed and the Leibniz product rule for the boundary operator is restored. Secondly, there is a $2h$ subcomplex introduced in Sullivan arXiv:1811.00086 and discussed further in Lawrence-Sullivan-Ranade arXiv:2011.07505 which shares the above good properties when ideal elements are added AND which also has a star bijection between degree zero and degree three and between one and degree two. This is introduced for the purposes of computations of 3D fluid motion, incompressible, with or without viscosity. For the latter purposes one only needs the good properties in dimensions zero, one and two, where the situation is a bit better. It is a new feature that the three good properties are respected by the crumbling chain mappings from coarse to finer subdivisions. The star operator does not cooperate with crumbling and is the sole reason in this discrete approximation for the Kolmogorov cascade to finer scales.