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 : 7

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :The inertia subgroup Ininsert ignore into journalissuearticles values(p); of a surgery obstruction group Lninsert ignore into journalissuearticles values(p); is generated by elements that act trivially on the set of homotopy triangulations Sinsert ignore into journalissuearticles values(X); for some closed topological manifold Xn-1 with p1insert ignore into journalissuearticles values(X); = p. This group is a subgroup of the group Cninsert ignore into journalissuearticles values(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 Jninsert ignore into journalissuearticles values(p); \subset Ininsert ignore into journalissuearticles values(p);, which is generated by elements of the group Lninsert ignore into journalissuearticles values(p);, which act trivially on the set S\partialinsert ignore into journalissuearticles values(X, \partial X); of homotopy triangulations relative to the boundary of any compact manifold with boundary insert ignore into journalissuearticles values(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 Lninsert ignore into journalissuearticles values(p);. In the present paper we describe all possible invariants that can give a Browder--Livesay filtration for computing the subgroup Jninsert ignore into journalissuearticles values(p);. These are invariants of elements x \in Lninsert ignore into journalissuearticles values(p);, which are nonzero if x \notin Jninsert ignore into journalissuearticles values(p);. More precisely, we prove that a Browder--Livesay filtration of a given manifold can give the following invariants of elements x \in Lninsert ignore into journalissuearticles values(p);, which are nonzero if x \notin Jninsert ignore into journalissuearticles values(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