Trinitary algebras

TL;DR

This paper classifies and analyzes the structure of trinitary algebras, which are subalgebras of smooth functions distinguished by conditions at three points, with applications to the study of singular geometric objects.

Contribution

It classifies trinitary algebras up to codimension four, computes their cohomology rings, and identifies characteristic classes, advancing understanding of their geometric and topological properties.

Findings

Classification of trinitary algebras up to codimension four.

Computation of cohomology rings of their varieties.

Identification of Stiefel-Whitney classes of normal bundles.

Abstract

A {\em $k$-trinitary algebra} is any subalgebra of the space of smooth functions $f: M \to {\mathbb R}$ that is distinguished in this space by $k$ independent conditions of the form $f(x_i) = f(\tilde x_i) = f(\hat x_i)$, where $x_i, \tilde x_i,$ and $ \hat x_i $ are distinct points in $ M$, $i=1, \dots, k$, or is approximated by such subalgebras. Trinitary algebras naturally arise in the study of {\em discriminant varieties,} that is, the spaces of singular geometric objects, when the property of being singular is formulated in terms of the simultaneous behavior at three distinct points. The simplest singular objects of this kind are the plane curves with triple self-intersections, see \cite{A}, \cite{MD}. The spaces of all $k$-trinitary algebras in $C^\infty(M, {\mathbb R})$ are analogous to the spaces of all ideals of finite codimension, which play the same role in the study of discriminants defined in the terms of a single singular point. These spaces are also analogous to the spaces of {\em equilevel algebras} (see \cite{EA}), which arise in the study of discriminants defined by binary singularities. We classify the trinitary algebras up to the codimension four in $C^\infty(S^1, {\mathbb R})$, compute the cohomology rings of their varieties and find the Stiefel--Whitney classes of their canonical normal bundles. We also present a series of $(2k-2)$-dimensional cohomology classes of the spaces of trinitary algebras of codimension $2k$ for any natural $k$.