Celebratio Mathematica — Birman

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J. S. Birman and J. Powell: “Special representations for 3-manifolds,” pp. 23–51 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537723 Zbl 0471.57002 incollection

Using techniques from the link calculus, it is shown that each closed, oriented 3-manifold admits a very special type of Heegaard decomposition which is also a very special surgery. The associated sewing map is the restriction to the Heegaard surface of a homeomorphism of a handlebody which fixes a complete system of meridian discs. The groups of isotopy classes of all such maps is closely related to the classical pure braid group. The associated Heegaard diagram consists of a pair of curves \( (\mathbf{x},\mathbf{y}) \) which have the following property: there is an associated diagram \( (\mathbf{x},\mathbf{z}) \) for \( S^3 \) such that \( (\mathbf{y},\mathbf{z}) \) also defines \( S^3 \) and the intersection matrices \( \mathbf{x} \cap \mathbf{z} \) and \( \mathbf{y} \cap \mathbf{z} \) are each the identity matrix. These diagrams have a great deal of interesting structure, including certain self-duality and finiteness properties.

James Cecil Cantrell	Related
Jerome E. Powell	Related