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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.