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

Friedrich Hegenbarth and Dušan Repovš

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 {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.

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
Friedrich Hegenbarth
Department of Mathematics
University of Milano
Via C Saldini 50
Milano
Italy 02130
http://www.mat.unimi.it/
Dušan Repovš
Institute for Mathematics, Physics and Mechanics
University of Ljubljana
Jadranska 19
Ljubljana
Slovenia 1001
http://pef.pef.uni-lj.si/~dusanr/index.htm