Universal $L^2$-torsion, polytopes and applications to $3$-manifolds

Abstract: Given an $L<sup>2$-acyclic connected finite $CW$-complex, we define its universal $L<sup>2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}<sup>w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincar\'e duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group $\operatorname{Wh}<sup>w(G)$ to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal $L<sup>2$-torsion can be identified with many invariants such as the $L<sup>2$-torsion, the $L<sup>2$-torsion function, twisted $L<sup>2$-Euler characteristics and, in the case of a $3$-manifold, the dual Thurston norm polytope.