Unbounded Gaps Between Ordinary and Equivariant Dehn Surgery Numbers

Abstract

For a closed oriented $3$-manifold $Y$ and an orientation-preserving involution $τ$, let $\DS(Y)$ denote the minimum number of components in an integral surgery description of $Y$, and let $\EDS(Y,τ)$ denote the corresponding minimum among periodic surgery descriptions inducing $τ$. We prove that for every integer $k\geq 1$ there is a pair $(Y_k,τ_k)$ such that \[ \DS(Y_k)=k, \qquad \EDS(Y_k,τ_k)=2k. \] Consequently, the difference $\EDS(Y,τ)-\DS(Y)$ is unbounded even when $τ$ is an involution. This answers Problems~1.15(b) and~1.15(c) in the K3 problem list. We also construct infinitely many pairwise nonhomeomorphic irreducible lens spaces $Z$ admitting involutions $σ$ for which \[ \DS(Z)<\EDS(Z,σ). \]