Surgery in codimension 3 and the Browder--Livesay invariants

Authors : Friedrich Hegenbarth, Yuri V. Muranov, Dusan Repovs

Pages : 806-817

Doi:10.3906/mat-1202-13

View : 16 | Download : 9

Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :The inertia subgroup In(p) of a surgery obstruction group Ln(p) is generated by elements that act trivially on the set of homotopy triangulations S(X) for some closed topological manifold Xn-1 with p1(X) = p. This group is a subgroup of the group Cn(p), 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 Jn(p) \subset In(p), which is generated by elements of the group Ln(p), 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 Ln(p). In the present paper we describe all possible invariants that can give a Browder--Livesay filtration for computing the subgroup Jn(p). These are invariants of elements x \in Ln(p), which are nonzero if x \notin Jn(p). More precisely, we prove that a Browder--Livesay filtration of a given manifold can give the following invariants of elements x \in Ln(p), which are nonzero if x \notin Jn(p): 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--Quin