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Surgery in codimension 3 and the Browder-Livesay invariants. (English) Zbl 1344.19002
Summary: 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 $$\mathcal S(X)$$ 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 $$\mathcal S^{\partial}(X, \partial X)$$ 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.
MSC:
 19J25 Surgery obstructions ($$K$$-theoretic aspects) 55T99 Spectral sequences in algebraic topology 58A35 Stratified sets 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)
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