[Paper Review] Involutory Hopf algebras and 3-manifold invariants

Greg Kuperberg|arXiv (Cornell University)|May 19, 1990

Algebraic structures and combinatorial models6 references4 citations

TL;DR

This paper constructs a 3-manifold invariant from any finite-dimensional involutory Hopf algebra using tensor network contractions on a Heegaard diagram or triangulation. The key result shows that the invariant counts homomorphisms from the fundamental group to a group G when the Hopf algebra is the group algebra of G, and establishes a one-to-one correspondence between formal tensor expressions and 3-manifolds modulo a known equivalence, suggesting a potential algorithm for 3-manifold homeomorphism detection.

ABSTRACT

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Motivation & Objective

• To define a topological invariant for 3-manifolds using finite-dimensional involutory Hopf algebras.
• To establish a correspondence between formal tensor expressions and 3-manifolds modulo a well-understood equivalence.
• To explore whether such invariants can lead to an algorithm for determining 3-manifold homeomorphism.
• To generalize the state-sum construction of 3-manifold invariants beyond quantum groups to broader algebraic structures.
• To provide a framework where the invariant is computable via tensor contractions on triangulations or Heegaard splittings.

Proposed method

• The invariant is defined using a state-sum model on a Heegaard diagram or triangulation of a 3-manifold.
• Tensor products and contractions of the structure tensors (multiplication, comultiplication, etc.) of the involutory Hopf algebra are used to compute the invariant.
• The construction is invariant under handle slides and stabilizations, ensuring topological invariance.
• The method relies on the algebraic properties of involutory Hopf algebras, particularly the antipode satisfying S² = id.
• A formal expression involving the algebra’s tensors corresponds uniquely to a 3-manifold up to a known equivalence relation.
• The framework allows for explicit computation of the invariant through combinatorial contraction of tensors on a cellular decomposition.

Experimental results

Research questions

• RQ1 Can a 3-manifold invariant be constructed from any finite-dimensional involutory Hopf algebra using tensor network contractions?
• RQ2 Does the invariant constructed from the group algebra of G count homomorphisms from the fundamental group of the 3-manifold to G?
• RQ3 Is there a one-to-one correspondence between formal tensor expressions built from the Hopf algebra and 3-manifolds modulo a well-understood equivalence?
• RQ4 Can this correspondence lead to an algorithm for deciding whether two 3-manifolds are homeomorphic?
• RQ5 How does the state-sum construction on Heegaard diagrams or triangulations relate to the topological invariance of the resulting invariant?

Key findings

• The invariant is well-defined and independent of the choice of Heegaard diagram or triangulation, due to invariance under handle slides and stabilizations.
• When the Hopf algebra is the group algebra of a finite group G, the invariant equals the number of group homomorphisms from the fundamental group of the 3-manifold to G.
• Every formal expression built from the structure tensors of the Hopf algebra corresponds to a unique 3-manifold modulo a known equivalence relation.
• The construction provides a computable state-sum model for 3-manifold invariants using only the algebraic data of an involutory Hopf algebra.
• The framework suggests a potential algorithmic approach to the 3-manifold homeomorphism problem via tensor expression equivalence.
• The method generalizes previous invariants based on quantum groups and extends the scope of topological quantum field theory constructions.