The Bryant-Ferry-Mio-Weinberger construction for generalized manifolds
Contributo in Atti di convegno
Data di Pubblicazione:
2006
Citazione:
The Bryant-Ferry-Mio-Weinberger construction
for generalized manifolds / F. Hegenbarth, D. Repovs - In: Exotic homology manifolds / [a cura di] Frank Quinn, Andrew Ranicki. - Coventry : Mathematics Institute, University of Warwick, 2006. - pp. 17-32 (( convegno Mini-Workshop exotic homology manifolds tenutosi a Oberwolfach nel 2003.
Abstract:
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.
Tipologia IRIS:
03 - Contributo in volume
Keywords:
generalized manifold, Poincaré duality, ε{−}δ–surgery, controlled, Quinn index, Poincaré complex, ANR, cell-like resolution
Elenco autori:
F. Hegenbarth, D. Repovs
Link alla scheda completa:
Titolo del libro:
Exotic homology manifolds