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

Luft, E.; Sjerve, D.

doi:10.1090/S0002-9947-1984-0748842-2

Involutions with isolated fixed points on orientable $3$-dimensional flat space forms
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by E. Luft and D. Sjerve PDF

Trans. Amer. Math. Soc. 285 (1984), 305-336 Request permission

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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