Geometry & Topology Monographs 9 (2006) 17–32

DOI: 10.2140/gtm.2006.9.17

arXiv: math/0608654

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 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 p_i+1: X_i+1→X_i. Our main theorem requires a sufficiently controlled Poincaré structure on X_i (over X_i-1). Our construction shows that this can be achieved. In fact, the Poincaré structure of X_i 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 X_i's has no resolution.

Keywords

generalized manifold, Poincaré duality, ε−δ–surgery, controlled, Quinn index, Poincaré complex, ANR, cell-like resolution

Mathematical Subject Classification

Primary: 57PXX

Secondary: 55RXX

References

Publication

Received: 7 July 2005
Accepted: 7 July 2005
Published: 22 April 2006

Authors