On 3-manifolds That Have Finite Fundamental Group and Contain Klein Bottles

The closed irreducible 3-manifolds with finite fundamental group and containing an embedded Klein bottle can be identified with certain Seifert fibre spaces. We calculate the isotopy classes of homeomorphisms of such 3-manifolds. Also we prove that a free involution acting on a manifold of this type, gives as quotient either a lens space or a manifold in this class. As a corollary it follows that a free action of Z8 or a generalized quaternionic group on S3 is equivalent to an orthogonal action. 0. Introduction. We are in the PL category. The object of study is the class of closed, irreducible orientable 3-manifolds which contain embedded Klein bottles and have finite fundamental group. These 3-manifolds are easily shown to be exactly the Seifert fibre spaces [7] with at most 3 exceptional fibres of multiplicity 2, 2, p (p > 1) and the 2-sphere as orbit surface, excluding S2 X S1. We prove that any homeomorphism homotopic to the identity is isotopic to the identity for such a 3-manifold M (this was done for a particular case where p = 2 in [4]). Also the factor group of the group of orientation-preserving homeomorphisms of M by the normal subgroup of homeomorphisms isotopic to the identity, which is denoted %(M), is shown to be one of the groups Z2, Z2 + Z2, S2 and S3 + Z2. There are no orientation-reversing homeomorphisms of M. Finally we establish that any free involution on M gives as quotient either a lens space or a 3-manifold in the above class. Let Q(8 m) be the group [x,y\x2 = (xyf = y2m}. As a corollary it follows that a free action of 0(2*) on S3, k > 3, is equivalent to an orthogonal action. Also simpler proofs of the analogous result in [5] and [6] for Z4 and Zg are given. Note that the 3-manifolds in the above class are not sufficiently large. Therefore it is interesting to see that some of the results of Waldhausen [9] can be achieved in this case. In another paper [11] we will build on the work here to obtain that free actions of some finite groups of order 2m3" onS3 are equivalent to orthogonal actions. Received by the editors November 24, 1976 and, in revised form, March 20, 1978. AMS (MOS) subject classifications (1970). Primary 57A10, 57E05, 57E25; Secondary 55A10.