The non-simply connected Price twist for the 4-sphere

TL;DR

This work analyzes non-simply connected Price twists $\tau_K$ along $P^2$-knots of Kinoshita type in $S^4$, showing these twists arise as pochette surgeries and connecting them to known 4-manifold families. A core result is that $\tau_K$ is the double of a 2-handlebody $F(K\#P_0)$ and, for ribbon 2-knots of 1-fusion, $\tau_K$ is diffeomorphic to $\tau_{S(T_{2,n})}$ with $n=\det(K)$, enabling a complete classification in this family. The authors develop a novel handle-diagram calculus (α-, β-deformations) to study diffeomorphism types, and exhibit examples where $\pi_1(\tau_K)$ is dihedral or other Coxeter groups, including explicit cases with $\Sigma_2(\tau_{S(T_{2,2n+1})})\cong L_{2n+1}\#S^2\times S^2$. By reformulating results via pochette surgery they relate Price twists to Iwase and Pao manifolds and demonstrate infinite families of non-diffeomorphic pochette surgeries, broadening the landscape of 4-manifold surgery classifications.

Abstract

A cutting and pasting operation on a $P^2$-knot $S$ in a $4$-manifold is called the Price twist. The Price twist for the $4$-sphere $S^4$ yields at most three $4$-manifolds up to diffeomorphism, namely, the $4$-sphere $S^4$, the other homotopy $4$-sphere $Σ_{S}(S^4)$ and a non-simply connected $4$-manifold $τ_{S}(S^4)$. In this paper, we study some properties and diffeomorphism types of $τ_{S}(S^4)$ for $P^2$-knots $S$ of Kinoshita type.

Figures (69)

• Figure 1: Handle diagrams of the Pao manifolds $L_n$$(\varepsilon=0)$ and $L_n'$$(\varepsilon=1)$.
• Figure 2: A simplified handle diagram of a $2$-handlebody $F(K\#P_0)$. For the definition of this diagram, see Section \ref{['sec:diffeo type']}.
• Figure 3: A handle diagram of $N(S)$ and three exceptional fibers $S_0$, $S_1$ and $S_{-1}$ in $\partial{N(S)}$ with normal Euler number $e(S)=\pm2$.
• Figure 4: A handle diagram of the exterior $E(K)$ of a 2-knot $K$ in $S^4$.
• Figure 5: A handle diagram of the exterior $E(K \# P_0^{\pm2})$ of a 2-knot $K$ and the unknotted $P^2$-knots $P_0^{\pm2}$ in $S^4$.

Theorems & Definitions (90)

• Proposition : Proposition \ref{['prop:pricepochette']}
• Corollary : Corollary \ref{['cor:homology']}
• Proposition : Proposition \ref{['prop:pao']}
• Theorem : Theorem \ref{['thm:pi1oftaut22n+1']}
• Corollary : Corollary \ref{['cor:not spin case']}
• Corollary : Corollary \ref{['cor:tauandPao']}
• Corollary : Corollary \ref{['cor:tauandIwasemfd']}
• Theorem : Theorem \ref{['thm:double']}
• Theorem : Theorem \ref{['thm:1-fusion']}
• Corollary : Corollary \ref{['cor:2-bridge']}