In geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the lantern relation is a relation that appears between certain Dehn twist

Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...

s in the mapping class group

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...

 of a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

. The most general version of the relation involves seven Dehn twists.

General form

The general form of the lantern relation involves seven Dehn twists in the mapping class group of a disk

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

 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

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

 (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

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic 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

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 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

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

, 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.

External links

• Sketches of Topology – The Lantern Relation