Compact n-Manifolds via (n + 1)-Colored Graphs: a New Approach

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, and it is dual to the one given by Pezzana in the 1970s. In this context we establish some results concerning the topology of the represented manifolds: suspensions, fundamental groups, connected sums and moves between graphs representing the same manifold. Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained, together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.