Surgery in codimension 3 and the Browder–Livesay invariants

Abstract

The inertia subgroup In (π) of a surgery obstruction group Ln (π) is generated by elements that act trivially on the set of homotopy triangulations S(X) for some closed topological manifold X n−1 with π1 (X) = π . This group is a subgroup of the group Cn (π) , which consists of the elements that can be realized by normal maps of closed manifolds. These 2 groups coincide by a recent result of Hambleton, at least for n ≥ 6 and in all known cases. In this paper we introduce a subgroup Jn (π) ⊂ In (π) , which is generated by elements of the group Ln (π) , which act trivially on the set S ∂ (X, ∂X) of homotopy triangulations relative to the boundary of any compact manifold with boundary (X, ∂X) . Every Browder–Livesay filtration of the manifold X provides a collection of higher-order Browder–Livesay invariants for any element x ∈ Ln (π) . In the present paper we describe all possible invariants that can give a Browder–Livesay filtration for computing the subgroup Jn (π) . These are invariants of elements x ∈ Ln (π) , which are nonzero if x ∈ Jn (π) . More / precisely, we prove that a Browder–Livesay filtration of a given manifold can give the following invariants of elements x ∈ Ln (π) , which are nonzero if x ∈ Jn (π) : the Browder-Livesay invariants in codimensions 0, 1, 2 and a class of / obstructions of the restriction of a normal map to a submanifold in codimension 3.