Additivity for parametrized topological Euler characteristic and   Reidemeister torsion
  Dwyer, Weiss, and Williams have recently defined the notions of parametrized topological Euler characteristic and parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and Reidemeister torsion of finite CW-complexes. 
Bernard Badzioch, Wojciech Dorabiala