Generalized handles in graphs and connected sums of manifolds

Abstract

Generalized handles in (n+1)-coloured graphs are defined. They are the n-dimensional analogue of a concept listed in [19] and generalizes the definition of combinatorial handle first introduced in [8] and successively investigated in[2],[12],[19],[26]. 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 Sn−1 -bundles over S1 as factors of a decomposition of a manifold in connected sum.