CONTROLLED SURGERY WITH TRIVIAL LOCAL FUNDAMENTAL GROUPS

Abstract

We provide a proof of the controlled surgery sequence, including stabil- ity, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasaki. In this note we provide a proof of the controlled surgery exact sequence used in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Wein- berger, (BFMW). A primitive version of controlled surgery was developed by the second author in his definition of the invariant that identifies exotic homology man- ifolds, (Q3,Q4). Surgery with bounded control, including exact sequences, was de- veloped in (FP). The remarkable limit construction of (BFMW) uses a refinement of the sequences of (FP). Roughly speaking (FP) describes a limit as ! 0 while (BFMW) depends crucially on a stability property of the limiting process. The proof of the refinement was postponed to a planned project that was never com- pleted. The intent was to deduce stability in general from a special case with an independent solution known to be stable, the � approximation theorem of (CF). This is reasonable in principle and may be possible, but it has become clear that the authors of (BFMW) have not addressed serious technical issues needed to actu- ally carry it out. As noted in the review (R3) Until (the planned project) or some substitute becomes available the surgery classification must be regarded as somewhat provisional - although there is little doubt among the experts that it is correct. The purpose of this paper is to provide the appropriate substitute. Our proof is direct, and is based on work of Yamasaki (Y). The following is Theorem 2.4 of (BFMW) with minor inaccuracies corrected. 1. Theorem. Suppose B is a finite dimensional compact metric ANR and a dimension n 4 is given. There is a stability threshold �0 > 0 so that for any �0 > > 0 there is > 0 with the following property: If f : N ! B is (�,1)- connected and N is a compact n-manifold then there is a controlled surgery exact sequence