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:

- The complex is connected and simply connected.
- Each
*edge*(one-cell) of is labelled by an arrow and a letter*a*∈*A*. - Some
*vertex*(zero-cell) which belongs to the topological boundary of is specified as a*base-vertex*. - 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 .

Read more about Van Kampen Diagram: Example, Van Kampen Lemma, Generalizations and Other Applications, See Also, Basic References

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