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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Involutions with isolated fixed points on orientable $ 3$-dimensional flat space forms


Authors: E. Luft and D. Sjerve
Journal: Trans. Amer. Math. Soc. 285 (1984), 305-336
MSC: Primary 57N10; Secondary 57S17, 57S25
MathSciNet review: 748842
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Abstract: In this paper we completely classify (up to conjugacy) all involutions $ \iota: M \to M$, where $ M$ is an orientable connected flat $ 3$-dimensional space form, such that $ \iota$ has fixed points but only finitely many. If $ M_1,\ldots,M_6$ are the $ 6$ space forms then only $ M_1, M_2, M_6$ admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori $ T \subseteq M$ so that either $ \iota(T) = T$ or $ \iota(T) \cap T = \varnothing$ and then cut $ M$ into simpler pieces. These results lead to a complete classification of $ 3$-manifolds containing $ \mathbf{Z} \oplus \mathbf{Z} \oplus \mathbf{Z}$ in their fundamental groups.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0748842-2
PII: S 0002-9947(1984)0748842-2
Article copyright: © Copyright 1984 American Mathematical Society