Irreducible triangulations of 2-manifolds with boundary

This is a talk on joint work with Maŕıa José Chávez, Seiya Negami, Antonio Quintero, and Maŕıa Trinidad Villar. A triangulation of a 2-manifold M is a simplicial 2-complex with underlying space homeomorphic to M . The operation of contraction of an edge e in a triangulation T of M is the operation which consists in contracting e to a single vertex and collapsing each face that meets e to a single edge. T is called an irreducible triangulations if none of the edges of T can be contracted without leaving the category of simplicial complexes; in other words, no edge of T can be contracted without either creating multiple edges or changing the topological type of the underlying space. In fact, in the case of a closed M other than the 2-sphere there is only one impediment to edge contractibility: (1) an edge is non-contractible if and only if it appears in more than two cycles of length 3 (made up of edges of T ). In case of a 2-manifold with boundary there are two additional impediments to contractibility—it is impossible to contract: (2) a chord (that is, a nonboundary edge with both end vertices in the boundary) and (3) a boundary edge if the length of the boundary is 3. Vertex splitting is the operation inverse to the edge contraction. Observe that all triangulations of M can be generated by repeatedly splitting the irreducible triangulations of M (see [1, 8, 10]). Irreducible triangulations have proved to be an effective tool for solving problems in combinatorial topology of 2-manifolds and discrete geome-