Surface bundles over surfaces: new inequalities between signature, simplicial volume and Euler characteristic

We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.Comment: 14 pages. Simplified the proof of Proposition 1.2. This is the final version, accepted in Geometriae Dedicat

Cohomology of symplectic groups and Meyer's signature theorem

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, \mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.Comment: 18 pages. Minor corrections. The most important one is in the table for $g=1$ on page 16: two columns had been swapped in the previous version. This is the version accepted for publication in Algebraic and Geometric Topolog

Surface bundles over surfaces: a study through signature and simplicial volume

La géométrie des fibrés en surface au-dessus du cercle a été étudiée en profondeur par Thurston, et plus récemment par Agol. Le cas suivant à considérer est celui des fibrés en surface au-dessus de surfaces. Ces 4-variétés sont porteuses de nombreuses questions ouvertes, telles l'existence d'une structure hyperbolique sur leur espace total. La présente thèse étudie les fibrés en surface au-dessus de surfaces au moyen d'invariants topologiques dont les principaux sont la signature et le volume simplicial. Ce dernier est un invariant dont peu de valeurs exactes sont connues à ce jour, mais qui présente l'avantage d'avoir des connexions fortes avec la géométrie des variétés. Ce travail se compose de deux directions de recherche différentes: la première étudie les fibrés en surface au-dessus de surfaces en combinant et en comparant les invariants à disposition. De nouvelles inégalités sont obtenues. La deuxième direction de recherche nous conduit à l'étude de la signature modulo 8 des fibrés en surface au-dessus de surfaces. Nous montrons l'existence de classes de cohomologie à coefficients réduits analogues à celles produites par Meyer et Turaev dans le cadre des entiers

INTEGRAL FOLIATED SIMPLICIAL VOLUME AND CIRCLE FOLIATIONS

We show that the integral foliated simplicial volume of a compact oriented smooth manifold with a regular foliation by circles vanishes

PRODUCTS OF FREE GROUPS IN LIE GROUPS

For every Lie group G, we compute the maximal n such that an n-fold product of nonabelian free groups embeds into G

Digital revolution equals digital competencies? What we expect for workers’ competencies in Industry 4.0

The chapter discusses the impact of Industry 4.0 technologies on workers’ need for competencies and skills, analyzing the main challenges imposed by digital revolution on soft skills combination and upgrade. It also proposes a cause for reflection on Italian government, universities and other stakeholders’ awareness and engagement in providing adequate frameworks, solutions and training to face the dynamic evolution of skills’ demand by the labour market. Its main contribution is that it provides insights on the state-of-the art of the match of supply with demand of soft skills in the Italian context and reflects upon desired trends for the near future