Surgery in codimension 3 and the Browder{Livesay invariants

The inertia subgroup I_n(pi) of a surgery obstruction group L_n(pi) is generated by elements that act trivially on the set of homotopy triangulations S(X) for some closed topological manifold X^{n-1} with pi_1(X) = pi. This group is a subgroup of the group C_n(pi), 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 geq 6 and in all known cases. In this paper we introduce a subgroup J_n(pi) subset I_n(pi), which is generated by elements of the group L_n(pi), which act trivially on the set S^{partial}(X, partial X) of homotopy triangulations relative to the boundary of any compact manifold with boundary (X, partial X). Every Browder--Livesay filtration of the manifold X provides a collection of higher-order Browder--Livesay invariants for any element x in L_n(pi). In the present paper we describe all possible invariants that can give a Browder--Livesay filtration for computing the subgroup J_n(pi). These are invariants of elements x in L_n(pi), which are nonzero if x notin J_n(pi). More precisely, we prove that a Browder--Livesay filtration of a given manifold can give the following invariants of elements x in L_n(pi), which are nonzero if x notin J_n(pi): 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.