The Bryant-Ferry-Mio-Weinberger construction of generalized manifolds

Abstract

Following Bryant, Ferry, Mio and Weinberger we construct generalized manifolds as limits of controlled sequences p_i: X_i --> X_{i-1} : i = 1,2,... of controlled Poincar\'e spaces. The basic ingredient is the epsilon-delta-surgery sequence recently proved by Pedersen, Quinn and Ranicki. Since one has to apply it not only in cases when the target is a manifold, but a controlled Poincar\'e complex, we explain this issue very roughly. Specifically, it is applied in the inductive step to construct the desired controlled homotopy equivalence p_{i+1}: X_{i+1} --> X_i. Our main theorem requires a sufficiently controlled Poincar\'e structure on X_i (over X_{i-1}). Our construction shows that this can be achieved. In fact, the Poincar\'e structure of X_i depends upon a homotopy equivalence used to glue two manifold pieces together (the rest is surgery theory leaving unaltered the Poincar\'e structure). It follows from the epsilon-delta-surgery sequence (more precisely from the Wall realization part) that this homotopy equivalence is sufficiently well controlled. In the final section we give additional explanation why the limit space of the X_i's has no resolution.