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Higher-Dimensional Generalized Manifolds: Surgery and Constructions
EMS Series of Lectures in Mathematics

Alberto Cavicchioli (Università degli Studi di Modena e Reggio Emilia, Italy)
Friedrich Hegenbarth (Università degli Studi di Milano, Italy)
Dušan Repovš (University of Ljubljana, Slovenia)

Higher-Dimensional Generalized Manifolds: Surgery and Constructions

ISBN print 978-3-03719-156-9, ISBN online 978-3-03719-656-4
DOI 10.4171/156
May 2016, 154 pages, softcover, 17 x 24 cm.
32.00 Euro

Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself.

A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful.

Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds.

This is the first book to systematically collect the most important material on higher-dimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology.

Keywords: Homology manifold, Poincaré duality, degree 1 normal map, boundedly controlled surgey, surgery spectrum, assembly map, Quinn index, Euclidean neighborhood retract, cell-like resolution, disjoint disks property, manifold recognition problem