J.H. Rubinstein, On 3-manifolds that have finite fundamental groups and contain Kleinian bottles, Trans. Amer. Math. Soc. 351 (1979), 129-137.  Home/Search   Document Not in Database   Summary   Related Articles

....T 3 , c) M is S 1 D 2 or a I bundle over the torus or Klein bottle. # It is now known that two homeomorphisms of Seifert fiber spaces are homotopic if and only if they are isotopic, see [Wa] for Haken manifolds, Sc2] for irreducible Seifert manifolds with infinite fundamental groups, and [BO, RB, HR, R, B, La] for various cases of Seifert manifolds covered by S 3 and S 2 S 1 . Thus the word homotopic in the theorem can be replaced by isotopic . The theorem says that if M is not one of the listed manifolds, then Seifert fibrations on M are unique up to isotopy. We will restrict our discussion ....

....two cone points of order 2. Then p 1 (#) is a Klein bottle K 1 in M . Note that if k = 2 then there are three cone points of HOMEOMORPHISMS OF 3 MANIFOLDS AND THE NIELSEN NUMBER 13 order 2, so there are three such arcs, and hence three Klein bottles K 1 , K 2 , K 3 . It was shown in the proof of [R, Theorem 6] that f can be deformed by an isotopy so that if k #= 2 then f maps K 1 to itself, and if k = 2 then f maps K 1 to one of the K i . Note that each K i is a union of fibers of M , so it gives a fibration of K i with orbifold an arc with two cone points. A regular fiber of K i is a separating ....

J.H. Rubinstein, On 3-manifolds that have finite fundamental groups and contain Kleinian bottles, Trans. Amer. Math. Soc. 351 (1979), 129-137.

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