Browder-Livesay Filtrations and the Example of Cappell and Shaneson

Abstract

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 \({\sigma (f) \in L_{5}(\pi _{1}(M))}\) is based on consideration of a Browder-Livesay filtration of a manifold X with \({\pi _{1}(X) \cong \pi _{1}(M)}\) . For a Browder-Livesay pair \({Y^{n-1} \subset X^{n}}\), 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 Γn. 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 4k+1 with \({\pi _{1}(X) \cong Q}\) . 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.