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Theorem 1 of this announcement constructs explicit algebraic counterexamples to the general conjecture (see for example [W, p. 138]) that groups of Hermitian forms satisfy a sum formula for free products. This conjecture was verified when the relevant groups have no 2-torsion by proving equivalent codimension one splitting theorems for manifolds. In part II of this note, see especially Theorem 7, the failure of the general algebraic conjecture leads to examples of nonsplittable manifolds with Z 2 <= n1. Some of the geometric splitting obstructions occur as differences of Arf-Kervaire invariants of base spaces and covering spaces.

Splitting obstructions for Hermitian forms and manifolds with 𝑍₂⊂𝜋₁