Surgery in codimension 3 and the Browder-Livesay invariants.

*(English)*Zbl 1344.19002Summary: 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) |