Generalized handles in graphs and connected sums of manifolds

We introduce the concept of generalized handles in (n+1)-colored graphs, and determine many combinatorial and topological properties of them. Then the operations of cancelling and cutting a generalized handle are studied. As a consequence, some decomposition theorems about manifolds are proved. In particular, a graph-theoretical condition is obtained to recognize S^(n-1)-bundles over S^1 as factors of a decomposition of a manifold in connected sum.