topological graph theory In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in ...

, the Wilson operations are a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of six transformations on

graph embedding In topological graph theory, an embedding (also spelled imbedding) of a graph G on a surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with vertices and simple arcs (homeomorphic images of ,1/math>) ...

s. They are generated by two involutions on embeddings, surface duality and

Petrie dual In topological graph theory, the Petrie dual of an graph embedding, embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces. The Petrie dual is also called th ...

ity, and have the group structure of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

on three elements. They are named for Stephen E. Wilson, who published them for regular maps in 1979; they were extended to all cellular graph embeddings (embeddings all of whose faces are topological disks) by . The operations are: identity, duality, Petrie duality, Petrie dual of dual, dual of Petrie dual, and dual of Petrie dual of dual or equivalently Petrie dual of dual of Petrie dual. Together they constitute the

S3. These operations are characterized algebraically as the only

outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...

s of certain group-theoretic representations of embedded graphs. Via their action on

dessins d'enfants In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for ...

, they can be used to study the

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...

of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s. One can also define corresponding operations on the edges of an embedded graph, the partial dual and partial Petrie dual, such that performing the same operation on all edges simultaneously is equivalent to taking the surface dual or Petrie dual. These operations generate a larger group, the ribbon group, acting on the embedded graphs. As an abstract group, it is isomorphic to

S_3^m

, the

m

-fold product of copies of the three-element symmetric group.

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