[Paper Review] Dehn Surgery Equivalence Relations on Three-Manifolds

Tim D. Cochran, Amir Gerges+1 more|arXiv (Cornell University)|Sep 28, 1998

Geometric and Algebraic Topology13 references4 citations

TL;DR

This paper investigates equivalence relations on 3-manifolds generated by Dehn surgeries that preserve first homology, showing that such relations are deeply tied to classical invariants like the fundamental group and the lower central series. It establishes connections to Heegaard splittings and the Torelli group, providing a foundational framework for finite-type invariants of 3-manifolds via surgery on restricted link types.

ABSTRACT

When can one 3-manifold be transformed to another by a finite sequence of Dehn surgeries which are restricted to preserve the first homology of the manifolds ? What is the resulting equivalence relation on 3-manifolds ? What if the surgery circle is further restricted to lie more deeply in the lower central series of the fundamental group ? We answer these questions. It is shown that many of these questions have answers in terms of classical toplogical invariants. Relations with Heegard splittings and the Torelli group are discussed. This is also related to whether or not one 3-manifold may be obtained from another by Dehn surgery on a link of restricted type. These equivalence relations form the philosophical basis of the authors joint work with Paul Melvin on a theory of finite type invariants for arbitrary 3-manifolds.

Motivation & Objective

• To determine when one 3-manifold can be transformed into another via finite sequences of Dehn surgeries that preserve the first homology group.
• To characterize the equivalence relation on 3-manifolds induced by such surgeries.
• To examine the impact of restricting surgery curves to deeper levels of the lower central series of the fundamental group.
• To explore connections between these equivalence relations and classical topological structures such as Heegaard splittings and the Torelli group.
• To provide a foundational framework for the theory of finite-type invariants of 3-manifolds, as developed in joint work with Paul Melvin.

Proposed method

• The authors analyze Dehn surgery operations on knots or links in 3-manifolds, focusing on surgeries that induce trivial action on the first homology group H₁.
• They employ algebraic topology tools, particularly the lower central series of the fundamental group, to restrict surgery curves to those lying deeper in the group’s nilpotent structure.
• The study uses homological and homotopical techniques to compare manifolds under surgery equivalence, emphasizing invariants preserved under such operations.
• The paper relates surgery equivalence to Heegaard splittings by analyzing how surgery affects Heegaard decompositions and their associated mapping class groups.
• It leverages the Torelli group as a key object to understand the group of automorphisms preserving homology and linking forms.
• Theoretical constructions and invariants such as the linking form and Reidemeister torsion are used to distinguish or relate manifolds under the equivalence.

Experimental results

Research questions

• RQ1 When can a 3-manifold be transformed into another via a finite sequence of Dehn surgeries that preserve the first homology group?
• RQ2 What is the structure of the equivalence relation on 3-manifolds generated by such surgeries?
• RQ3 How does restricting surgery curves to lie in deeper levels of the lower central series of the fundamental group affect the resulting equivalence classes?
• RQ4 What is the relationship between surgery equivalence and Heegaard splittings of 3-manifolds?
• RQ5 How do these equivalence relations relate to the theory of finite-type invariants for 3-manifolds?

Key findings

• The equivalence relation generated by Dehn surgeries preserving H₁ is characterized by classical invariants such as the linking form and the first homology group.
• Surgery on curves in the second term of the lower central series of π₁(M) preserves the homology and leads to a well-defined equivalence relation on 3-manifolds.
• The Torelli group acts as a natural symmetry group for the surgery equivalence relation, reflecting automorphisms that act trivially on homology.
• The equivalence relation is compatible with Heegaard splittings, and surgery operations can be understood in terms of handlebody modifications.
• The results provide a topological foundation for the theory of finite-type invariants of 3-manifolds, particularly in the context of surgery on algebraically restricted links.
• The paper establishes that surgery equivalence under these constraints is deeply connected to the structure of the fundamental group and its nilpotent quotients.