Obstructions for generalized graphmanifolds to be nonpositively curved

Abstract

An $n$-dimensional manifold $M$ ($n\ge 3$) is called {\it generalized graph manifold} if it is glued of blocks that are trivial bundles of $(n-2)$-tori over compact surfaces (of negative Euler characteristic) with boundary. In this paper two obstructions for generalized graph manifold to be nonpositively curved are described. Each 3-dimensional generalized graph manifold with boundary carries a metric of nonpositive sectional curvature in which the boundary is flat and geodesic (B. Leeb). The last part of this paper contains an example of 4-dimensional generalized graph manifold with boundary, which does not admit a metric of nonpositive sectional curvature with flat and geodesic boundary.