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The $ \mu $-invariant of $ 3$-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented $ 2$-manifold


Authors: Joan S. Birman and R. Craggs
Journal: Trans. Amer. Math. Soc. 237 (1978), 283-309
MSC: Primary 57A10
DOI: https://doi.org/10.1090/S0002-9947-1978-0482765-9
MathSciNet review: 0482765
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Abstract: Let $ \mathcal{H}(n)$ be the group of orientation-preserving selfhomeomorphisms of a closed oriented surface Bd U of genus n, and let $ \mathcal{K}(n)$ be the subgroup of those elements which induce the identity on $ {H_1}({\text{Bd}}\;U;{\mathbf{Z}})$. To each element $ h \in \mathcal{H}(n)$ we associate a 3-manifold $ M(h)$ which is defined by a Heegaard splitting. It is shown that for each $ h \in \mathcal{H}(n)$ there is a representation $ \rho $ of $ \mathcal{K}(n)$ into $ {\mathbf{Z}}/2{\mathbf{Z}}$ such that if $ k \in \mathcal{K}(n)$, then the $ \mu $-invariant $ \mu (M(h))$ is equal to the $ \mu $-invariant $ \mu (M(kh))$ if and only if k $ \in $ kernel $ \rho $. Thus, properties of the 4-manifolds 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{K}(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.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0482765-9
Article copyright: © Copyright 1978 American Mathematical Society

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