# Hegenbarth Friedrich, Kolokvij september 2016

### Iz MaFiRaWiki

The notion of generalized manifolds (g.m.'s) has passed through various stages. The most common contemporary definition is characterized by two properties: (i) every g.m. *X* is an ENR-space and (ii) every g.m. *X* has the same local homology groups as the Euclidean *n*-space , where *n* is the dimension of *X*.

One advantage of these two properties is that the Poincare duality can be expressed in terms of the singular (co-)homology instead of the more complicated sheaf-theoretical setting. Moreover, *X* has the homotopy type of a Poincare duality complex in the sense of Wall (as follows from the classical Borsuk Conjecture, proved by West).

A milestone in the development of the theory of g.m.'s was the Edwards DDP Theorem which states that if a g.m. *X*^{n} of dimension admits a resolution and satisfies the disjoint disk property (DDP) then *X* is homeomorphic to a topological *n*-manifold.

To construct resolutions, i.e. cell-like maps , where *M* is an *n*-manifold, Quinn invented controlled surgery techniques. Surprisingly, there is an integer obstruction *i*(*X*) for the existence of a resolution. This integer *i*(*X*) is part of a surgery obstruction, and the Wall-realization of such obstructions were used by Bryant-Ferry-Mio-Weinberger to systematically construct g.m.'s.

A striking property of *i*(*X*) is its local character: If *U* is an open subset of *X*, then *i*(*U*) = *i*(*X*). This might not hold for earlier notions of g.m.'s. Other properties of the invariant *i*(*X*) will be discussed in the lecture, following the presentation in our new monograph A. Cavicchioli, F. Hegenbarth and D. Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions, European Mathematical Society, Zurich, 2016.