geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the lantern relation is a

relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

that appears between certain

Dehn twist In geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area dis ...

s in the

mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...

of a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

. The most general version of the relation involves seven Dehn twists. The relation was discovered by Dennis Johnson in 1979.

General form

The general form of the lantern relation involves seven Dehn twists in the mapping class group of a disk with three holes, as shown in the figure on the right. According to the relation, : where , , and are the right-handed Dehn twists around the blue curves , , and , and , , , are the right-handed Dehn twists around the four red curves. Note that the Dehn twists , , , on the right-hand side all commute (since the curves are disjoint, so the order in which they appear does not matter. However, the

cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Ins ...

of the three Dehn twists on the left does matter: : Also, note that the equalities written above are actually equality up to

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

or isotopy, as is usual in the mapping class group.

General surfaces

Though we have stated the lantern relation for a disk with three holes, the relation appears in the mapping class group of any surface in which such a disk can be embedded in a nontrivial way. Depending on the setting, some of the Dehn twists appearing in the lantern relation may be homotopic to the

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

, in which case the relation involves fewer than seven Dehn twists. The lantern relation is used in several different presentations for the mapping class groups of surfaces.

References

External links

Sketches of Topology – The Lantern Relation
Geometric topology Homeomorphisms {{topology-stub