Über den Rand von Homologiemannigfaltigkeiten (Q1088208)

A homology manifold (over k) is a finite connected simplicial complex with relative homology groups (over k) of a sphere. For such a complex \(\Delta\) define Bd \(\Delta\) :\(=(\sigma \in \Delta:\) \(H_ N(\delta,\cos t_{\Delta}\sigma;k)=0)\) \((N=\dim \Delta)\). This subcomplex is the boundary of \(\Delta\) and coincides with the usual boundary if \(\Delta\) is a finite triangulation of a compact topological manifold. The main result states that this boundary is a (N-1)-dimensional homology manifold without boundary provided \(\Delta\) is orientable (over k). This generalizes a result of the author [Math. Nachr. 117, 161-174 (1984; Zbl 0586.13011)]. If \(\Delta\) is not orientable (over a field k) the boundary can behave ''wild''. E.g., the cone over the projective plane has a boundary consisting of the plane itself and the vertex of the cone (char \(k\neq 2)\). In the case \(k={\mathbb{Z}}\), Bd \(\Delta\) will be a (N-1)- dimensional quasimanifold (ibid, p. 165) without boundary even if \(\Delta\) is not orientable. It remains open whether Bd \(\Delta\) must be a homology manifold in this case.