Integrality in topology stemming from complex algebraic varieties

Tóm tắt báo cáo: A  smooth complex quasi-projective variety $X$ is, as a topological space,  homotopic to a bouquet of spheres.  In particular its fundamental group $\pi_1^{\rm top}(X)$  is finitely presented, thus finitely generated and the relations themselves are finitely generated. For example, if $X$ is an affine curve, $\pi_1^{\rm top}(X)$ is a free group. Free groups possess linear  integral representations into $GL_r(r, \bar{\mathbb{Z}})$ for all $r$. Not every finitely presented group possesses  linear  integral representations. So we could ask whether this property is an obstruction for a finitely presented group to come from geometry, that is to be isomorphic to $\pi_1^{\rm top}(X)$ for some $X$ as above. While we have at disposal obstructions which stem from Hodge theory, such as harmonic analysis, when $X$ is projective (or proper), they do not apply for $X$ quasi-projective.