Hegenbarth Friedrich, Kolokvij september 2016

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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 \mathbb{R}^{n}, 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. Xn of dimension \ge 4 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 f:M^{n}\to X^{n}, 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.

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