Grant Cairns

Chaotic actions of free groups

An action of a group, G, on a space, M, is chaotic if it is topologically transitive and the set of points with finite orbit is a dense subset of M. In this paper we show that every compact triangulable manifold of dimension greater than one admits a faithful chaotic action of every countably generated free group.

NEW BOUNDS ON THE BETTI NUMBERS OF NILPOTENT LIE ALGEBRAS

We provide new upper and lower bounds for the Betti numbers of nilpotent Lie algebras. As an application, we prove the toral rank conjecture (TRC) for nilmanifolds of dimension at most 14 and for a small class of coformal spaces. We also give a new, direct proof of the result of Deninger and Singhof that the TRC is true for 2-step nilpotent Lie algebras.

Explicit Betti numbers for a family of nilpotent Lie algebras

Betti numbers for the Heisenberg Lie algebras were calculated by Santharoubane in his 1983 paper. However few other examples have appeared in the literature. In this note we give the Betti numbers for a family of (2n+1)dimensional 2-step nilpotent extensions of R by R2n. Introduction Let g denote a finite dimensional nilpotent Lie algebra defined over an arbitrary field k. Let g∗ denote the vector space dual to g and ∧g∗ = ⊕ i≥0 ∧ig∗ the exterior algebra. The differential d : ∧g∗ → ∧g∗ is the unique derivation of degree one extending dx∗(a ∧ b) = −x∗([a, b]) for each x∗ ∈ ∧1g∗ and a, b ∈ g. We calculate the Betti numbers bi(g) of g given by bi(g) = dim(H (g, k)) for the Lie algebra cohomology with coefficients in k. For every n ∈ N, let hn denote the n Heisenberg Lie algebra. This is the (2n+1)-dimensional 2-step nilpotent Lie algebra with basis {x1, . . . , xn, y1, . . . , yn, z} and non-zero relations [xi, yi] = z for each 1 ≤ i ≤ n. According to Santharoubane [5] bi(hn) = ( 2n i ) − ( 2n i− 2 ) for all 0 ≤ i ≤ n (assuming ( p q ) = 0 unless 0 ≤ q ≤ p). The remaining numbers are given by Poincaré duality. Recall that the Heisenberg Lie algebras arise as extensions of R by R. We study a family of (2n+1)-dimensional 2-step nilpotent extensions of R by R. For every n ∈ N, let gn denote the Lie algebra with basis {x1, . . . , xn, y1, . . . , yn, z} and non-zero relations [z, xi] = yi for each 1 ≤ i ≤ n. Our main result is the following. Received by the editors April 20, 1994 and, in revised form, August 31, 1995. 1991 Mathematics Subject Classification. Primary 17B56; Secondary 17B30, 22E40.

Generalized Thrackle Drawings of Non-bipartite Graphs

A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.

The Crystallographic Restriction, Permutations, and Goldbach's Conjecture

1. INTRODUCTION. The object of this paper is to make an observation connecting Goldbachs conjecture, the crystallographic restriction, and the orders of the elements of the symmetric group. First recall that for an element g of a group G the order Ord(g) of g is defined to be the smallest natural number such that g

A new formula for winding number

We give a new formula for the winding number of smooth planar curves and show how this can be generalized to curves on closed orientable surfaces. This gives a geometric interpretation of the notion of winding number due to B. Reinhart and D.R.J. Chillingworth.

On the Betti Numbers of Nilpotent Lie Algebras of Small Dimension

The work of Golod and Safarevic on class field towers motivated the conjecture that b2 > b2 1/4 for nilpotent Lie algebras of dimension at least 3, where b i denotes the i th Betti number. Using a new lower bound for b 2 and a characterization of Lie algebras of the form g/Z(g), we prove this conjecture for 2-step algebras. We also give the Betti numbers of nilpotent Lie algebras of dimension at most 7 and use them to establish the conjecture for all nilpotent Lie algebras whose centres have codimension ≤ 7.

FURTHER GEOMETRY OF THE MEAN CURVATURE ONE-FORM AND THE NORMAL PLANE FIELD ONE-FORM ON A FOLIATED RIEMANNIAN MANIFOLD

For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form K and the normal plane field one-form /S. Through examples, we show that an important result of Kamber-Tondeur on K is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

Outerplanar Thrackles

We show that a graph drawing is an outerplanar thrackle if and only if, up to an inversion in the plane, it is Reidemeister equivalent to an odd musquash. This establishes Conway’s thrackle conjecture for outerplanar thrackles. We also extend this result in two directions. First, we show that no pair of vertices of an outerplanar thrackle can be joined by an edge in such a way that the resulting graph drawing is a thrackle. Secondly, we introduce the notion of crossing rank; drawings with crossing rank 0 are generalizations of outerplanar drawings. We show that all thrackles of crossing rank 0 are outerplanar. We also introduce the notion of an alternating cycle drawing, and we show that a thrackled cycle is alternating if and only if it is outerplanar.

The answer to Woodall's musquash problem

Abstract We prove that there are no n -agonal musquashes for n even with n≠6 . This resolves a problem raised in Woodalls 1971 paper ‘Thrackles and Deadlock’.