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The propose of this paper is to define numerical invariants of odd-dimensional surgery obstructions, computable in a way similar to that used to compute the index and Arf invariants of even-dimensional surgery obstructions. The main result is that a system of integral congruences ("numerical invariants") suffices, modulo the projective class group, to determine whether or not an odd-dimensional surgery obstruction vanishes, when the f undumental group is a finite 2-group. In addition, the numerical invariants turn out to be Euler characteristics in certain cases of topological interest, including the existence of product formulas.

The exact sequence of a localization for Witt groups. II. Numerical invariants of odd-dimensional surgery obstructions