$3$-manifolds that are sums of solid tori and Seifert fiber spaces

Abstract

It is s1Aown that a simply connected 3-manifold is S3 if it is a sum of a Seifert fiber space and solid tori. Let F be an orientable Seifert fiber space with a disk as orbit surface. It is shown that a sum of F and a solid torus is a Seifert fiber space or a connected sum of lens spaces. Let M be a closed 3-manifold which is a union of three solid tori. It is shown that M is a Seifert fiber space or the connected sum of two lens spaces (including S1 x S2). Let M be a sum of a Seifert fiber space and solid tori. It is shown (Theorem 1) that if M is simply connected, then M is S3. This generalizes Hempel's result in [4]. Let F be a Seifert fiber space with orbit surface a disk. A particular example is the complement (of a regular neighborhood in S3) of a torus knot. The structure of all 3 manifolds N that are a sum of F and a solid torus is described (Theorem 3). Also the structure of all 3-manifolds that are a union of three solid tori (such that the intersection of any two is an annulus) is described (Theorem 4). In particular a question of A. C. Connor [3] is answered in the affirmative, that any such 3-manifold is S3 if it is simply connected. 1. We work throughout in the piecewise linear category. A sum of a 3-manifold F and solid tori V1, * * *, Vn is the manifold obtained from F and V1, * * *, V, by identifying components Ti of aFwith aVi under homeomorphisms fi: aViTi (i= 1, * , n). The connected sumn M1 #M2 of two 3-manifolds is the manifold obtained by removing 3-balls in int(M1) and int(M2), and identifying the resulting 2-sphere boundaries under an orientation reversing homeomorphism. For a definition and classification of Seifert fiber spaces, see [5] and [6]. If F is a Seifert fiber space there is a map P of F onto its orbit surface f (Zerlegungsflache). The image of an exceptional fiber is an exceptional point onf Received by the editors March 10, 1972 and, in revised form, May 9, 1972. AMS (MOS) subject classifications (1970). Primary 57A10, 55A40.