The combinatorial transverse intersection algebra
Daniel An, Ruth Lawrence, Dennis Sullivan
TL;DR
The paper develops a combinatorial transverse intersection algebra on the $h$-cubical model of the 3-torus by enlarging the chain complex with infinitesimal elements to restore the boundary product rule. It proves a Comprehensive Theorem establishing a locally defined transverse intersection product on $EC_*$ that is graded-commutative, associative, and Leibniz-compatible, matching geometric intersection in general position and compatible with lattice symmetries, while furnishing a nondegenerate Frobenius-type pairing under odd periodicity. The construction is built from the 1D case via tensor products, yields a uniquely determined subalgebra $FC_*$ where the product holds, and extends to arbitrary dimensions with an Addendum prescribing truncation limits $m = \left\lfloor\dfrac{2n}{3}\right\rfloor$. These results provide a finite, computable framework for discretized intersection theory with potential applications to 3D fluid dynamics and cascade analyses through discrete approximations. All results are expressed with careful attention to geometric and combinatorial consistency, including compatibility with crumbling maps and symmetry constraints.
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.