The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here “compute” means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet).Next,...

We describe explicitly the group of transverse diffeomorphisms of several types of minimal linear foliations on the torus $T^n$, $n\ge 2$. We show in particular that non-quadratic foliations are rigid, in the sense that their only transverse diffeomorphisms are $\pm \mathrm{Id}$ and translations. The description derives from a general formula valid for the group of transverse diffeomorphisms of any minimal Lie foliation on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and...

Let $A\subset {ℝ}^n$ be a set-germ at $0\in {ℝ}^n$ such that $0\in \overline{A}$. We say that $r\in S^{n-1}$ is a direction of $A$ at $0\in {ℝ}^n$ if there is a sequence of points $\left\{x_i\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^n$ such that $\frac{x_i}{\parallel x_i\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^n$.Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular if two subanalytic...

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on ${ℝ}^{2n}\times S^1$ and $ℝP^{2n+1}$. On the other hand we also show by elementary arguments that the...

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either $ℝ$ or $\mathrm{s}l(2,ℝ)$ or $so\left(3\right)$ are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to $n+1$, where $n$ is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For $ℝ$-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf...