Van Kampen Diagram

A van Kampen diagram over the presentation (†) is a planar finite cell complex, given with a specific embedding with the following additional data and satisfying the following additional properties:

1. The complex is connected and simply connected.
2. Each edge (one-cell) of is labelled by an arrow and a letter a∈A.
3. Some vertex (zero-cell) which belongs to the topological boundary of is specified as a base-vertex.
4. For each region (two-cell) of for every vertex the boundary cycle of that region and for each of the two choices of direction (clockwise or counter-clockwise) the label of the boundary cycle of the region read from that vertex and in that direction is a freely reduced word in F(A) that belongs to R_∗.

Thus the 1-skeleton of is a finite connected planar graph Γ embedded in and the two-cells of are precisely the bounded complementary regions for this graph.

By the choice of R_∗ Condition 4 is equivalent to requiring that for each region of there is some boundary vertex of that region and some choice of direction (clockwise or counter-clockwise) such that the boundary label of the region read from that vertex and in that direction is freely reduced and belongs to R.

A van Kampen diagram also has the boundary cycle, denoted, which is an edge-path in the graph Γ corresponding to going around once in the clockwise direction along the boundary of the unbounded complementary region of Γ, starting and ending at the base-vertex of . The label of that boundary cycle is a word w in the alphabet A ∪ A−1 (which is not necessarily freely reduced) that is called the boundary label of .