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All $2$–dimensional links in $4$–space live inside a universal $3$–dimensional polyhedron

Cherry Kearton and Vitaliy Kurlin

Algebraic & Geometric Topology 8 (2008) 1223–1247

The hexabasic book is the cone of the 1–dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3–dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any closed 2–dimensional surface in 4–space is isotopic to a surface in UP. The proof is based on a representation of surfaces in 4–space by marked graphs, links with double intersections in 3–space. We construct a finitely presented semigroup whose central elements uniquely encode all isotopy classes of 2–dimensional surfaces.

2-knot, 2-link, handle decomposition, hexabasic book, marked graph, singular link, universal polyhedron, 3-page book, 3-page embedding, universal semigroup
Mathematical Subject Classification 2000
Primary: 57Q45, 57Q35, 57Q37
Received: 7 April 2008
Revised: 7 June 2008
Accepted: 13 June 2008
Published: 26 July 2008
Cherry Kearton
Department of Mathematical Sciences, Durham University
Durham DH1 3LE
United Kingdom
Vitaliy Kurlin
Department of Mathematical Sciences, Durham University
Durham DH1 3LE
United Kingdom