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A SPECTRAL SEQUENCE IN SURGERY

By I. Khmblton [i. Hambleton and A. F. Kharshiladze

Abstract

ABSTRACT. A spectral sequence whose differentials are connected with iterated Browder-Livesay invariants is constructed. Some examples are considered. Bibliography: 16 titles. It has been known for some time that iterated Browder-Livesay invariants resemble the differentials in a spectral sequence (see [2], [3], [4], [7], [8], [14]). S. Cappell and J. Shaneson in their paper [3] asked: what is this "spectral sequence"? In this paper we provide an answer by identifying the iterated Browder-Livesay invariants with the differentials of the homotopy spectral sequence of a suitable filtration on /--theory and begin to investigate its properties. Recall that if G is a group with given orientation character w: G —> {±1} , and π c G is a subgroup of index 2, then the Browder-Livesay groups LNn (π — » G) are defined (by W. Browder and G. Livesay in [1] for π = 1, and in general by C. T. C. Wall in [11]). These groups appear as the obstruction groups for one-sided submanifold splitting problems. Let Υ be a manifold with πι (Υ) = G and W\(Y) = w. Consider a mapping φ: Υ —> RP N, Ν> 2 dim Υ, which induces an epimorphism of fundamental groups and is transversal to the projective subspace RP N ~ i C RP N. Let \χτ{φ,: π {Υ) ^ Ζ/2} = π λ and denote by X = φ ~ 1 {ΚΡ Ν ~ 1) the induced one-sided codimension 1 submanifold. Assume that η = dimX> 4 and that the inclusion X c Υ induces an isomorphism of fundamental groups, as one can arrange by deforming the map φ. If /: Ν —> Υ is a simple homotopy equivalence of closed manifolds, an obstruction in the group LNn{n —> G) is defined, which is zero if and only if the map / is splittable along X c Υ. If Ν, Υ have boundaries assume that / is split along the boundary, and then the obstruction is also defined. There are two natural maps, which provide a connection between the Browder-Livesay groups and the Wall groups Ln(G): and c: LN n(n- G)-> L n(G~) Here G ~ is the same as G as a group, but the orientation character of G ~ is defined by w<f>, where φ: G — • Z/2 is the epimorphism with kernel π. (i) If χ e LN (n — » G) is the splitting obstruction for some homotopy equivalenc

Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.188.7913
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