geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated i ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 *Public ...

that appears between certain Dehn twists in the mapping class group of a surface. 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 Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...

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 Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

(since the curves are disjoint, so the order in which they appear does not matter. However, the cyclic order 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, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

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, un ...

, 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