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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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
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by Joan S. Birman and R. Craggs PDF
Trans. Amer. Math. Soc. 237 (1978), 283-309 Request permission

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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Additional Information
  • © Copyright 1978 American Mathematical Society
  • 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