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Given an    L  2   ‐acyclic connected finite    C  W   ‐complex, we define its universal    L  2   ‐torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead groupWh  w    (  G  )    . We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead groupWh  w    (  G  )to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal    L  2   ‐torsion can be identified with many invariants such as the    L  2   ‐torsion, the    L  2   ‐torsion function, twisted    L  2   ‐Euler characteristics and, in the case of a 3‐manifold, the dual Thurston norm polytope.

Universal L 2 ‐torsion, polytopes and applications to 3‐manifolds