Consider a closed, orientable, irreducible 3-manifold M with |"i(M)| < oo, in which a Klein bottle can be embedded. We present a classification of the spaces M and show that, if irx{M) is cyclic, then M is homeomorphic to a lens space. Note that all surfaces of even genus can be embedded in each space M. We also classify all free involutions on lens spaces whose orbit spaces contain Klein bottles. 0. Introduction. Let M = M(K) be a closed, orientable, irreducible 3- manifold with ^(A/)! < oo, in which a Klein bottle K can be embedded. A large class of lens spaces belongs to the spaces M(K) (see Bredon and Wood (1)). The goal of this paper is to classify the spaces M(K), and investigate the relations between the spaces M(K) and lens spaces L(p, q). Especially, we extract the following result: Theorem 1. IfTTx(M) is abelian, then M is homeomorphic to a lens space. Throughout the paper we work in the PL category. We divide the paper into five sections. In §1 we define a 3-manifold M(p, q) for each pair (p, q) of relatively prime integers, and classify the spaces M (p, q). In §2 we classify the spaces M(K) by showing that each M(K) is homeomorphic to a space M(p, q) for some/?, q. In §3 we prove Theorem 1 and in §4 we investigate all free involutions on lens spaces whose orbit spaces contain Klein bottles. Our approach can be applied to an outstanding problem in the study of involutions on lens spaces. It has been asked by J. L. Tollefson whether each lens space L has a Heegaard splitting (L, F) of genus 1 (i.e., F splits L into two solid tori) such that F is invariant under a given involution h on the space L and P is in general position with respect to the fixed-point set Fix(A). All known involutions on lens spaces L have the property (that L has such a Heegaard splitting). In §5, we give new examples of involutions h with Fix(/i) ^0 which do not have the property. J. H. Rubinstein (personal correspondence with the author) has obtained a classification of the spaces M (K) independently and simultaneously.
[1]
Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich
,
1939
.
[2]
Yoko Tao.
On fixed point free involutions of $S\sp1\times S\sp2$
,
1962
.
[3]
Paik Kee Kim,et al.
PL involutions of fibered 3-manifolds
,
1977
.
[4]
K. W. Kwun,et al.
EXTENDING A PL INVOLUTION OF THE INTERIOR OF A COMPACT MANIFOLD.
,
1977
.
[5]
John W. Wood,et al.
Non-orientable surfaces in orientable 3-manifolds
,
1969
.
[6]
K. W. Kwun.
Scarcity of orientation-reversing ${\rm PL}$ involutions of lens spaces.
,
1970
.
[7]
C. D. Papakyriakopoulos,et al.
On Dehn's Lemma and the Asphericity of Knots
,
1957
.
[8]
J. Stallings.
On fibering certain 3-manifolds
,
1961
.
[9]
Friedhelm Waldhausen,et al.
On irreducible 3-manifolds which are sufficiently large *
,
2010
.
[10]
K. W. Kwun.
Sense-preserving ${\rm PL}$ involutions of some lens spaces.
,
1973
.
[11]
Paik Kee Kim.
involutions on lens spaces and other 3-manifolds
,
1974
.
[12]
J. Stallings.
On the Loop Theorem
,
1960
.