Trinitary algebras

TL;DR

The paper develops the theory of trinitary algebras, subalgebras of $C^ty(M,\mathbb{R})$ defined by triple-point jet conditions, and connects them to discriminant varieties and singular plane curves. It provides a complete classification of codimension-four trinitary algebras on $S^1$, builds a CW-structure for the corresponding moduli space $\overline{TD}_2(S^1)$, and computes its mod 2 cohomology and Stiefel–Whitney classes of the canonical normal bundle $\mathcal{N}_2$. A key contribution is the introduction of higher-codimension families via loops in the jet-parameter space, giving nontrivial $w_{2k-2}(\mathcal{N}_k)$ classes for all $k$ by constructing cycles $\Xi(a,b,c)$. The results bridge discriminant geometry, knot-theoretic diagrammatics, and finite-type invariants, with explicit incidence formulas and a detailed analysis of boundary maps, enabling higher-dimensional generalizations of trinitary phenomena. The work establishes foundational topological invariants for spaces of trinitary algebras and furnishes a framework for studying higher-codimension discriminants through algebraic-analytic limits.

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 $k$.

Figures (24)

• Figure 1: Series $A$
• Figure 2: $C$
• Figure 3: $B, E$
• Figure 4: $D$
• Figure 5: $F$

Theorems & Definitions (42)

• Theorem 1
• Definition 1
• Theorem 2
• Corollary 3
• Theorem 4
• Proposition 5
• Definition 2
• Lemma 6
• Theorem 7
• Proposition 8