Celebratio Mathematica — Birman

Interview with Joan Birman

Works connected to Robert Francis Craggs

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J. S. Birman and R. Craggs: “The \( \mu \)-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold,” Trans. Am. Math. Soc. 237 (March 1978), pp. 283–309. MR 0482765 Zbl 0383.57006 article

Let \( \mathcal{H}(n) \) be the group of orientation-preserving self-homeomorphisms of a closed oriented surface \( \operatorname{Bd} U \) of genus \( n \), and let \( \mathcal{H}(n) \) be the subgroup of those elements which induce the identity on \( H_1(\operatorname{Bd} U;\mathbf{Z}) \). To each element \( h \in \mathcal{H}(n) \) we associate a 3-manifold \( M(h) \) which is defined by a Heegard splitting. It is shown that for each \( h\in\mathcal{H}(n) \) there is a representation \( \rho \) of \( \mathcal{H}(n) \) into \( \mathbf{Z}/2\mathbf{Z} \) such that if \( k\in\mathcal{H}(n) \), then the \( \mu \)-invariant \( \mu(M(h)) \) is equal to the \( \mu \)-invariant \( \mu(M(kh)) \) if and only if \( k\in\operatorname{kernel} \rho \). Thus, properties of the 4-maniolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from \( \mathcal{H}(n) \) onto \( \mathbf{Z}/2\mathbf{Z} \) are studied and are shown to constitute a complete conjugacy class of subgroups of \( \mathcal{H}(n) \). The class has nontrivial finite order.