Browder-Livesay Filtrations and the Example of Cappell and Shaneson

Let M 3 be a 3-dimensional manifold with fundamental group π 1 ( M ) which contains a quaternion subgroup Q of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map f : M 3 × T 2 → M 3 × S 2 which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of is based on consideration of a Browder-Livesay filtration of a manifold X with . For a Browder-Livesay pair , the restriction of a normal map to the submanifold Y is given by a partial multivalued map Γ : L n ( π 1 ( X )) → L n−1 ( π 1 ( Y )), and the Browder-Livesay filtration provides an iteration Γ n . This map is a basic step in the definition of the iterated Browder-Livesay invariants which give obstructions to realization of surgery obstructions by normal maps of closed manifolds. In the present paper we prove that Γ 3 ( σ ( f )) = 0 for any Browder-Livesay filtration of a manifold X 4 k +1 with . We compute splitting obstruction groups for various inclusions ρ → Q of index 2, describe natural maps in the braids of exact sequences, and make more precise several results about surgery obstruction groups of the group Q .