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

Following Bryant, Ferry, Mio andWeinberger we construct generalized manifolds as limits of controlled sequences {Xi pi →Xi−1 | i = 1, 2, ...} of controlled Poincaré spaces. The basic ingredient is the ε-δ–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 Poincaré complex, we explain this issue very roughly (Theorem 3.5). Specifically, it is applied in the inductive step to construct the desired controlled homotopy equivalence pi+1 : Xi+1 → Xi . Theorem 3.5 requires a sufficiently controlled Poincaré structure on Xi (over Xi−1 ). Our construction shows that this can be achieved. In fact, the Poincaré structure of Xi depends upon a homotopy equivalence used to glue two manifold pieces together (the rest is surgery theory leaving unaltered the Poincaré structure). It follows from the ε-δ–surgery sequence (more precisely from the Wall realization part) that this homotopy equivalence is sufficiently well controlled. In §4 we give additional explanation why the limit space of the Xi ’s has no resolution. AMS Classification Primary 57PXX; Secondary 55RXX