On the universal pairing for 2-complexes

TL;DR

The paper extends the notion of the universal pairing from manifolds to 2-complexes and higher-dimensional complexes, proving that positivity fails in this setting. It shows that for 2-complexes, there exist nonzero elements x with x · x = 0, implying that unitary and semisimple TQFTs cannot distinguish stable equivalence from 3-deformations, paralleling cork phenomena in 4-manifolds. A refined pairing addressing simple homotopy via s-moves yields the same non-detectability, linking to the Andrews-Curtis conjecture. The results generalize to higher dimensions, demonstrating non-positivity persists for n ≥ 3 and suggesting a broad obstruction to positivity across dimensions. Together, these findings illuminate the limits of universal pairings as invariants for distinguishing homotopy-theoretic and deformation-type equivalences.

Abstract

The universal pairing for manifolds was defined and shown to lack positivity in dimension 4 by Freedman et al. We prove an analogous result for 2-complexes, and also show that the universal pairing does not detect the difference between simple homotopy equivalence and 3-deformations. The question of whether these two equivalence relations are different for 2-complexes is the subject of the Andrews-Curtis conjecture. We also discuss the universal pairing for higher-dimensional complexes and show that it is not positive.

Figures (2)

• Figure 1: The setting for a Quinn $s$-move in the smallest non-trivial example: a connected, genus $1$ surface $\Sigma$ with boundary $R\cup S$ and a symplectic basis of curves $a, b$. Right: the annulus $A$ is attached to $\Sigma$ along the curves $a, R$ and the annulus $B$ is attached to $\Sigma$ along $b, S$. The union $\Sigma\cup A\cup B$ is mapped to some $2$-complex $K$.
• Figure 2: Left: a null-homotopy for $RS^{-1}$ in $\Sigma\cup A\cup D_R$ is provided by the surface $\Sigma$ surgered along the disk $A\cup D_R$ attached to $a$. (The curve $RS^{-1}$ is defined using the induced orientation on the boundary of the surface $\Sigma$.) Right: a null-homotopy for $RS^{-1}$ in $\Sigma\cup B\cup D_S$ is given by $\Sigma$ surgered along the disk $B\cup D_S$ attached to $b$.

Theorems & Definitions (30)

• Remark 2.1
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Theorem 4.1
• proof : Proof of Theorem \ref{['thm: simple homotopy versus 3 deformations']}
• Lemma 4.2
• proof : Proof of Lemma \ref{['lem: can fix boundary']}
• Lemma 4.3
• proof : Proof of Lemma \ref{['lem: L_1.L_2']}