Topological and geometric properties of graph-manifolds

Details Topological and geometric properties of graph-manifolds Topological and geometric properties of graph-manifolds

2003-09-11

arxiv.org/abs/math/0309192v1

Mathematics

Geometric Topology

65 pages; a survey article. To appear in SPb Math. J

Scientific paper

This is an exposition of results on the existence problem of $\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are represented from a unified point of view based on the notion of compatible cohomological classes and some difference equation on the graph of a graph-manifold (BKN-equation). Criteria for seven different properties of graph-manifolds at three levels are given: at the level of compatible cohomological classes; at the level of solutions to the BKN-equation; in terms of spectral properties of operator invariants of a graph-manifold.

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