Wolfgang Heil

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

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 Hempels 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.

On the existence of incompressible surfaces in certain 3-manifolds.

If M is the closure of the complement of a regular neighborhood of a nontrivial knot in S3 then there exists a nonsingular torus T embedded in M, which is incompressible (i.e. the inclusion i: T->M induces a monomorphism i*: 7r1(T)-7r1(M)). If F is any orientable closed incompressible surface embedded in M then ri(M) contains ri(F) as a subgroup. L. Neuwirth [3, Question T] asks whether the converse is true: If 7rw(M) contains the group a of a closed (orientable) surface of genus g>1, does there exist a nonsingular closed surface F of genus g whose fundamental group is injected monomorphically into 7rw(M) by inclusion? As a partial answer we show that not for every such Cw7ri(M) there exists an incompressible FCM. The question remains open whether M contains incompressible closed surfaces of genus > 1. We show that for torus knots M does not contain such surfaces, by showing that 7ri(M) does not contain subgroups ~.

Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies

Abstract Every closed nanorientable 3-manifold M can be obtained as the union of three orientable handlebodies V1, V2, V3 whose interiors are pairwise disjoint. If gi denotes the genus of Vi, g1⩽g2⩽g3, we say that M has tri-genus (g1, g2, g3), if in terms of lexicographical ordering, the triple (g1, g2, g3) is minimal among all such decompositions of M into orientable handlebodies. We relate the tri-genus of M to the genus of a surface that represents the dual of the first Stiefel-Whitney class of M. This is used to determine g1 and g2.

COVERINGS OF 3-MANIFOLDS BY THREE OPEN SOLID TORI

Motivated by the concept of S1-category of a manifold, we obtain a classification of all closed 3-manifolds that can be covered by three open solid tori.

SEIFERT UNIONS AND SPACES OF GRAPHS IN S3

Let Γ be a non-splittable in S3, not a tree. It is shown that if the space of Γ is a Seifert union of solid tori then M is homeomorphic to the space of a tree of Burde-Murasugi links.

A Test for admissibility for two-dimensional, continuous fields on planar surfaces

Radar scatterometers on oceanic satellites provide wind stress data over the oceans except for a 180^o ambiguity in direction. Data sets consist of sticks, 2-dimensional vectors of known amplitude and direction, but undetermined orientation. In [1], it was shown that, for admissible stick fields, the assignment of an orientation to a single stick determines a unique realization of a vector field from the given continuous stick field. In this note, a criterion for determining admissibility of a given continuous stick field is given. This criterion depends only on the boundary curves of the region on which the stick field is given and it can be effectively checked.

Products of commutators and squares in free groups

Abstract Geometric proofs, using surface topology, are given for an upper bound of the rank of a subgroup determined by a quadratic equation in a free group.