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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57S17, 57S25

Retrieve articles in all journals with MSC: 57N10, 57S17, 57S25

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society