The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser–Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco–Shalen–Johannson decomposition of $M$.

In this paper we construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in ${\mathbb P}^2$. Such a pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over ${\mathbb Q}(\sqrt{5})$.

A coordinate-system called $\lambda$-lengths is constructed for an SL$(2,{\Bbb C})$ representation space of punctured surface groups. These $\lambda$-lengths can be considered as complexification of R. C. Penner's $\lambda$-lengths for decorated Teichmüller spaces of punctured surfaces. Via the coordinates the mapping class group acts on the representation space as a group of rational transformations. This fact is applied to find hyperbolic 3-manifolds which fibre over the circle.