Controlled Surgery with Good Local Fundamental Groups

The aim of this talk is to discuss a possibility to extend the following controlled surgery exact sequence: Theorem [PQR] (simplified version) Suppose B is a finite dimensional compact metric ANR, and a dimension n ≥ 4 is given. Then there exists a number 0> 0 which depends on B and n so that for any 0> > 0 there is δ> 0 with the following property: If p: X → B is UV 1 and X is a closed topological n-manifold then there is a controlled surgery exact sequence Hn+1(B,L) → S,δ(X, p) → [X,G/TOP] → Hn(B,L). For n ≥ 5, it seems that the above should hold true for reasonably good control maps (e.g. stratified systems of fibrations) p: X → B, if one replaces the homology groupsHi(B,L) with the controlled L-groups Lci (B, p). This may be obvious for experts, but not for me. I will really appreciate it if someone can help me writing down the detailed proof of the controlled surgery exact sequence in this generality. The reason we have homology when p is UV 1 is that the controlled Whitehead group Whc(B, p) vanihes for such p, and this in turn comes from the fact that the ordinar