Algebraic Presentation of $4$-Dimensional $2$-Handlebodies and $3$-Dimensional Cobordisms

In this paper, we give a new direct proof of a result by Bobtcheva and Piergallini that provides finite algebraic presentations of two categories, denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of dimension $3$ and $4$, respectively. More precisely, $3\mathrm{Cob}$ is the category of connected oriented $3$-dimensional cobordisms between connected surfaces with connected boundary, while $4\mathrm{HB}$ is the category of connected oriented $4$-dimensional $2$-handlebodies up to $2$-deformations. For this purpose, we explicitly construct the inverse of the functor $Φ: 4\mathrm{Alg} \to 4\mathrm{HB}$, where $4\mathrm{Alg}$ denotes the free monoidal category generated by a Bobtcheva–Piergallini Hopf algebra. As an application, we deduce an algebraic presentation of $3\mathrm{Cob}$ and show that it is equivalent to the one conjectured by Habiro.

💡 Research Summary

The paper provides a new direct proof of the finite algebraic presentations of two low‑dimensional topological categories: the category 4HB of connected oriented 4‑dimensional 2‑handlebodies up to 2‑deformation, and the category 3Cob of connected oriented 3‑dimensional cobordisms between connected surfaces with a single boundary component. The authors construct explicitly the inverse of the functor Φ from the free braided monoidal PRO 4Alg, generated by a Bobtcheva‑Piergallini (BP) Hopf algebra, to 4HB (or equivalently to the Kirby‑tangle category 4KT).

A BP Hopf algebra is a braided Hopf algebra equipped with an integral form λ, an integral element Λ, an invertible ribbon element τ, and a copairing w, satisfying a list of axioms (see Subsection 3.3). The free PRO 4Alg is generated by a single object carrying this structure; its morphisms are built from the usual Hopf operations (product, coproduct, unit, counit, antipode) together with the extra BP data.

The functor Φ assigns to each generator of 4Alg a standard Kirby tangle: the product and coproduct become 1‑handle attachments, the ribbon element becomes a full twist of a 2‑handle, etc. The authors verify that all defining relations of 4Alg hold in 4KT, establishing that Φ is a well‑defined braided monoidal functor.

The main technical contribution is the construction of the inverse functor Φ⁻¹: 4KT → 4Alg. For a given Kirby tangle T, the authors introduce a “bi‑ascending state”, i.e. a choice of crossings whose reversal trivializes the undotted link representing the 2‑handles. By connecting each undotted component to the base plane with a family of bands, they produce a diagram T_α in which every undotted component is doubled by a trivial copy lying underneath. This doubled diagram can be interpreted uniquely as a morphism in 4Alg. The paper proves that the resulting morphism is independent of the auxiliary choices (state, bands, ordering), using a careful analysis of band moves, handle slides, and the BP relations. Consequently Φ is shown to be full, faithful, and essentially surjective; hence an equivalence of braided monoidal categories.

An immediate corollary is an algebraic presentation of 3Cob. The boundary functor ∂⁺: 4HB → 3Cob induces a commutative diagram with Φ, and by imposing two additional relations on 4Alg (the factorizable and anomaly‑free conditions) one obtains a quotient category 3Alg. The authors prove that ∂⁺ ∘ Φ restricts to an equivalence 3Alg ≅ 3Cob. This recovers the presentation originally announced by Habiro and later proved by Kerler, and establishes the Kerler–Habiro conjecture: the Habiro Hopf algebra presentation 3Algᴴ and the BP‑based presentation 3Alg are equivalent as braided monoidal categories.

The paper also discusses the broader context. The 2‑deformation equivalence on 4‑handlebodies is conjectured by Gompf to be strictly weaker than diffeomorphism; detecting this difference would require a unimodular ribbon Hopf algebra that is non‑factorizable and non‑semisimple. Existing examples give only homological refinements of known 3‑dimensional quantum invariants. Moreover, the authors point out a symmetric monoidal quotient of 4Alg generated by a commutative BP Hopf algebra with trivial ribbon element; this category models 2‑dimensional CW‑complexes up to 2‑equivalence and is intended for studying the Andrews–Curtis conjecture.

In summary, the paper achieves three major goals: (1) a direct, constructive proof that the free BP‑Hopf‑algebra PRO 4Alg is equivalent to the topological category of 4‑dimensional 2‑handlebodies; (2) an explicit algebraic presentation of the 3‑dimensional cobordism category 3Cob, together with a proof of equivalence with Habiro’s presentation; and (3) a framework linking these algebraic structures to quantum invariants, TQFTs, and open problems such as the Andrews–Curtis conjecture and the distinction between diffeomorphism and 2‑deformation in dimension four.