Surgery in codimension 3 and the Browder--Livesay invariants


Abstract: 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.

Keywords: Surgery assembly map, closed manifolds surgery problem, assembly map, inertia subgroup, splitting problem, Browder--Livesay invariants, Browder--Livesay groups, normal maps, iterated Browder--Livesay invariants, manifold with filtration, Browder--Quinn surgery obstruction groups, elements of the second type of a Wall group

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