Triangulations of Seifert fibred manifolds

It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3-manifold in question is a Seifert fibred space.Revised version: 5 March 2004     

Dimensional linear spaces whose automorphism group is (line,hyperplane)-flag transitive

We classify the pairs (S, G) where S is a finite n-dimensional linear space with n 4 and G is an automorphism group of S acting transitively on the (line, hyperplane)-flags. We show in particular that S must be either a Desarguesian projective or affine space provided with its subspaces of dimension n - 1, or a Mathieu-Witt design provided with its blocks and its subsets of size n - 1. Our proof uses a recent classification of the flag transitive 2-(v, k, 1) designs, which in turn heavily depends on the classification of all finite simple groups. The case n = 3 has been settled in another paper.     

Algebras with transitive automorphism groups

Presented by I. Rosenberg.     

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study the modificationAA of an affine domainA which produces another affine domainA=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. First appeared in passing in the basic paper of O. Zariski [Zar], it was further considered by E. D. Davis [Da]. In [Ka1] its geometric counterpart was applied to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions (more general than those in [Ka1]) which guarantee preservation of the topology under a modification.As an application, we show that the group of biregular automorphisms of the affine hypersurfaceXC ^k+2, given by the equationuv=(p(x ₁,...,x_k) wherepC[x ₁,...,x _k ],k2, actsm-transitively on the smooth part regX ofX for anymN. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Partially supported by the NSA grant MDA904-96-01-0012.     

Inequivalent fibred knots whose homotopy Seifert pairings are isometric

Partially supported by a Northeastern University Jonior Research Fellowship     

Seifert fibrations of lens spaces

We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers \(\alpha _1^0,\alpha _2^0\), we give an algorithmic construction of a Seifert fibration \(L(p,q)\rightarrow S^2(\alpha |\alpha _1^0|,\alpha |\alpha _2^0|)\), where the natural number \(\alpha \) is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models.     

Doubly transitive automorphism groups of block designs

If G is a doubly transitive group of automorphisms of a block design with λ = 1, then for any block Δ of the design and any point α in Δ, the set Δ?{α} is a block of imprimitivity for G_α. What are sufficient conditions for a doubly transitive but not doubly primitive permutation group G to be a group of automorphisms of a non-trivial block design with λ = 1 ? Can the design or the group G be identified if there is a nonidentity automorphism in G fixing every point of some block of the design? Both of these questions are investigated and some answers are given.     

Non-commutative spaces with transitive translation groups

Let G be a finite group. The problem of finding all strongly skewaffine spaces with transitive translation group isomorphic to G will be reduced to the determination of all Schur rings over G.Dedicated to Helmut Salzmann on his 60th birthday     

Graphs with symmetric automorphism group

We investigate the properties of graphs whose automorphism group is the symmetric group. In particular, we characterize graphs on less than 2n points with group S_n, and construct all graphs on n + 3 points with group S_n. Graphs with 2n or more points and group S_n are discussed briefly.     

Profinite rigidity for Seifert fibre spaces

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.     

Heegaard splittings of Seifert fibered spaces

Summary In this paper we give a classification theorem of genus two Heegaard splittings of Seifert fibered manifolds overS ² with three exceptional fibers, except for when two of the exceptional fibers hava the same invariants with opposite orientation.