Symmetrically Bordered Surfaces

Abstract

We address the question: given a compact topological surface with boundary, when can it be symmetrically embedded in ℝ³? We construct examples of symmetric embeddings for compact surfaces with an odd number of boundary components, and connected sums of an even number of ℝℙ²'s (cross surfaces) with any number of boundary components. We show in the appendix that a connected sum of an odd number of ℝℙ²'s with 4n boundaries cannot be symmetrically embedded.

Additional information

Notes on contributors

William Cavendish

WILLIAM CAVENDISH is a graduate student at Princeton University. His research interests include geometric topology, hyperbolic geometry, and Teichmüller theory.

John H. Conway

JOHN H. CONWAY is the John von Neumann Professor of Mathematics at Princeton University.