Simon A. King

The computation of the cohomology rings of all groups of order 128

Abstract We describe the computation of the mod-2 cohomology rings of all 2328 groups of order 128. One consequence is that all groups of order less than 256 satisfy the strong form of Bensons Regularity Conjecture.

How to make a triangulation of S3 polytopal

We introduce a numerical isomorphism invariant p(T) for any triangulation T of S 3 . Although its definition is purely topological (inspired by the bridge number of knots), p(T) reflects the geometric properties of T. Specifically, if T is polytopal or shellable, then p(T) is small in the sense that we obtain a linear upper bound for p(T) in the number n = n(T) of tetrahedra of T. Conversely, if p(T) is small, then T is almost polytopal, since we show how to transform T into a polytopal triangulation by O((p(T)) 2 ) local subdivisions. The minimal number of local subdivisions needed to transform T into a polytopal triangulation is at least p(T)/3n - n - 2. Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for p(T) exponential in n 2 . We prove here by explicit constructions that there is no general subexponential upper bound for p(T) in n. Thus, we obtain triangulations that are very far from being polytopal. Our results yield a recognition algorithm for S 3 that is conceptually simpler, although somewhat slower, than the famous Rubinstein-Thompson algorithm.

Minimal generating sets of non-modular invariant rings of finite groups

It is a classical problem to compute a minimal set of invariant polynomials generating the invariant ring of a finite group as a sub-algebra. We present here a new algorithmic solution in the non-modular case. Our algorithm only involves very basic operations and is based on well-known ideas. In contrast to the algorithm of Kemper and Steel, it does not rely on the computation of primary and (irreducible) secondary invariants. In contrast to the algorithm of Thiery, it is not restricted to permutation representations. With the first implementation of our algorithm in Singular, we obtained minimal generating sets for the natural permutation action of the cyclic groups of order up to 12 in characteristic 0 and of order up to 15 for finite fields. This was far out of reach for implementations of previously described algorithms. By now our algorithm has also been implemented in Magma. As a by-product, we obtain a new algorithm for the computation of irreducible secondary invariants that, in contrast to previously studied algorithms, does not involve a computation of all reducible secondary invariants.

The size of triangulations supporting a given link

LetT be a triangulation of S 3 containing a link L in its 1{skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces.

The Topological Representation of Oriented Matroids

Abstract We present a new proof of the Topological Representation Theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schonflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

Crossing number of links formed by edges of a triangulation

We study the crossing number of links that are formed by edges of a triangulation of S3 with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n2. In general, this bound can not be replaced by a subexponential bound. However, if is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant , called polytopality, introduced by the author.

Persistent homology of groups

Abstract We introduce and investigate notions of persistent homology for p-groups and for coclass trees of p-groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p-groups of order at most 81. The strength of these invariants, together with some of their elementary theoretical properties, suggest that persistent homology may be a useful tool in the study of prime-power groups. In particular, we ask whether the known periodic structure on coclass trees is reflected in a periodic structure on the persistent homology of p-groups in the trees.

The mod-2 cohomology ring of the third Conway group is Cohen-Macaulay

By explicit machine computation we obtain the mod-2 cohomology ring of the third Conway group Co_3. It is Cohen-Macaulay, has dimension 4, and is detected on the maximal elementary abelian 2-subgroups.

Complexity of triangulations of the projective space

Abstract It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3-dimensional projective space, in terms of the number of tetrahedra.

Regular flip equivalence of surface triangulations

Abstract Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that in general N is bigger than the minimal number of vertices of a triangulation. The existence of N was known, but no estimate. This paper provides an estimate for N that is linear in the Euler characteristic of the surface.