Volume 9 (2006)

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Exotic homology manifolds (Oberwolfach 2003)

Editors: Frank Quinn and Andrew Ranicki

[Group photograph]

Group photograph

This volume is the proceedings of the Mini-Workshop Exotic Homology manifolds held at Oberwolfach 29th June - 5th July, 2003. Homology manifolds were developed in the first half of the 20th century to give a precise setting for Poincaré's ideas on duality. Major results in the second half of the century came from two different areas. Methods from the point-set tradition were used to study homology manifolds obtained by dividing genuine manifolds by families of contractible subsets. `Exotic' homology manifolds are ones that cannot be obtained in this way, and these have been investigated using algebraic and geometric methods.

The Mini-Workshop brought together experts from the point-set and algebraic traditions, along with new PhD's and people in related areas. There were 17 participants, 14 formal lectures and a problem session. There was a particular focus on the proof of the existence of exotic homology manifolds. This gave experts in each area an the opportunity to learn more about details coming from other areas. There had also been concerns about the stability (`shrinking') theorem that in retrospect is a crucial step in the proof but had not been worked out when the theorem was originally announced. This was discussed in detail. One of the high points of the conference was the discovery of a short and very general new proof of this result by Pedersen and Yamasaki (published in these proceedings), so there are now three independent treatments. Extensive discussions of examples and problems clarified the current state of the field and mapped out objectives for the next decade.

A Mini-Workshop on history entitled `Henri Poincaré and topology' was held during the same week. There was joint discussion of the early history of manifolds, and each group offered evening lectures on topics of interest to the other. Several of the daytime history lectures also drew large numbers of homology manifold participants. The interaction between the two groups was very beneficial and should serve as a model for future such synergies.

We are grateful to the Oberwolfach Mathematics Institute for hosting the meeting, and to the participants, authors and the referees for their contributions.

Frank Quinn and Andrew Ranicki, August 2005

Geometry & Topology Monographs 9 (2006)

DOI: 10.2140/gtm.2006.9

Homologically arc-homogeneous ENRs

J L Bryant

Path concordances as detectors of codimension-one manifold factors

Robert J Daverman and Denise Halverson

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

Friedrich Hegenbarth and Dušan Repovš

The quadratic form E8 and exotic homology manifolds

Washington Mio and Andrew Ranicki

Stability in controlled L–theory

Erik Kjær Pedersen and Masayuki Yamasaki

Problems on homology manifolds

Frank Quinn

Controlled L–theory

Andrew Ranicki and Masayuki Yamasaki