Year: 2026
Type: preprint
Abstract: We show that there is no algorithm to decide whether or not a closed, connected, orientable, locally flat PL surface in $S^4$ of genus at least 18 (resp. 19) is TOP (resp. PL) unknotted. We prove analogous results for non-orientable surfaces, and for surfaces in any compact PL 4-manifold. We also show that there is an orientable surface in $S^4$ of genus 18 whose group has unsolvable word problem, and an orientable surface in $S^4$ of genus 40 whose group contains an isomorphic copy of every recursively presented group. Similar results hold for non-orientable surfaces.
Source: arXiv (Cornell University)
Author: Gordon, Cameron
Institution: The University of Texas at Austin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224133587