Following Bryant, Ferry, Mio and Weinberger we construct
generalized manifolds as limits of controlled sequences {pi:
Xi→ 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. Specifically,
it is applied in the inductive step to construct the desired controlled
homotopy equivalence pi+1: Xi+1→Xi.
Our main theorem 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 the final section we give additional explanation
why the limit space of the Xi's has no resolution.
