A survey of recent results on generalized manifolds (Q5936521)

A generalized \(n\)-manifold is a locally compact Euclidean neighbourhood retract \(X\) such that for any \(x\in X\), \(H_k(X,X-\{x\};\mathbb{Z})\) is equal to \(\mathbb{Z}\) if \(k=n\), and is equal to 0 otherwise. The paper is a survey of some recent results on the geometric topology of generalized manifolds, mainly in dimensions \(\geq 5\). The results concern the neighborhood classification, embeddings, and the concept of transversality of submanifolds. A clue notion is a so-called disjoint disks property (DDP) which states that every pair of maps of the 2-cell into \(X\) can be approximated by maps with disjoint images. The presented theorems can be interpreted as evidence that there is no essential difference between embedding and transversality theorems for topological manifolds and generalized manifolds that verify the DDP provided the codimension is at least three.