Cyclic surgery theorem¶

A three-manifold theorem bounding the distance between two Dehn fillings of an irreducible boundary-torus manifold when both filled manifolds have cyclic fundamental group.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4013
Origin domain: low dimensional topology
Subdomain: low dimensional topology

Core Idea¶

For a compact orientable irreducible three-manifold with torus boundary that is not a Seifert-fibered exceptional case, two cyclic surgery slopes have geometric intersection distance at most one. Character varieties and Culler–Shalen seminorms translate filling relations into algebraic curves and norm constraints, forcing distinct cyclic slopes to be adjacent. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Cyclic surgery theorem belongs to low dimensional topology and is useful where the analyst can specify the typed low dimensional topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate manifold compactness, orientability, irreducibility, torus boundary, non-Seifert and non-solid-torus hypotheses, slopes, Dehn filling convention, cyclic fundamental groups, and distance conclusion are explicit. The scope is broad within that domain but bounded by the need for manifold compactness, orientability, irreducibility, torus boundary, non-Seifert and non-solid-torus hypotheses, slopes, Dehn filling convention, cyclic fundamental groups, and distance conclusion are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making manifold compactness, orientability, irreducibility, torus boundary, non-Seifert and non-solid-torus hypotheses, slopes, Dehn filling convention, cyclic fundamental groups, and distance conclusion are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Cyclic surgery theorem can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Cyclic surgery theorem. Cyclic surgery theorem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed low dimensional topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express manifold compactness, orientability, irreducibility, torus boundary, non-Seifert and non-solid-torus hypotheses, slopes, Dehn filling convention, cyclic fundamental groups, and distance conclusion are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of low dimensional topology because they reuse the typed low dimensional topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Character varieties and Culler–Shalen seminorms translate filling relations into algebraic curves and norm constraints, forcing distinct cyclic slopes to be adjacent., and type the carrier, state every parameter and convention in the definition, test that manifold compactness, orientability, irreducibility, torus boundary, non-Seifert and non-solid-torus hypotheses, slopes, Dehn filling convention, cyclic fundamental groups, and distance conclusion are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Cyclic surgery theorem Domain-specific

Parents (1) — more general patterns this builds on

Cyclic surgery theorem is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Cyclic surgery theorem → Constraint

Neighborhood in Abstraction Space¶

Cyclic surgery theorem sits in a crowded region of the domain-specific corpus (14^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Topology & Homology (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08