Knotted surfaces, Homological Norm and Extendable Subgroup

TL;DR

This work addresses the problem of controlling extendable self-homeomorphisms for knotted surfaces in $S^4$ by introducing a Thurston-like norm on $H_1(K,\mathbb{Z})$ defined via the exterior of the knotted surface. By proving the additivity of this norm under connected sum and applying it to build high-genus embeddings $F_g\subset S^4$, the authors constrain the homological action of extendable maps to diagonal $\pm1$ matrices in $\mathrm{Sp}(2g,\mathbb{Z})$, with the torus case $g=1$ yielding finite image in $\mathrm{Aut}(T^2,\mathbb{Z})$. The core result shows that the image of the extendable group $E(i)$ on homology can be made small and highly structured, revealing a homological obstruction to extendability and suggesting finite-image phenomena for broader classes. The paper also discusses limitations in distinguishing mapping class group elements beyond homology and poses conjectures about Torelli-affected cases and potential higher-dimensional extensions.

Abstract

We prove that for arbitrary g, there is a surface K of genus g embedded in S4, which has finitely many extendable self-homeomorphisms' action on H1(K,Z), by defining a norm on H1(K,Z) and proving its additivity.

Figures (1)

• Figure 1: surgery

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 3.1
• Remark 3.2
• proof