In
mathematics, a pair of pants is 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 ...
which is
homeomorphic to the three-holed
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
. The name comes from considering one of the removed
disks as the waist and the two others as the cuffs of a
pair of pants
Trousers (British English), slacks, or pants are an item of clothing worn from the waist to anywhere between the knees and the ankles, covering both legs separately (rather than with cloth extending across both legs as in robes, skirts, and dr ...
.
Pairs of pants are used as building blocks for
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
surfaces in various theories. Two important applications are to
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, where decompositions of
closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
s into pairs of pants are used to construct the
Fenchel-Nielsen coordinates on
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...
, and in
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
where they are the simplest non-trivial
cobordisms between 1-dimensional
manifolds.
Pants and pants decomposition
Pants as topological surfaces
A pair of pants is any surface that is homeomorphic to a sphere with three holes, which formally is the result of removing from the sphere three
open disk
In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.
For a radius, r, an open disk is usu ...
s with pairwise disjoint closures. Thus a pair of pants is a compact surface of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
zero with three
boundary components.
The
Euler characteristic of a pair of pants is equal to −1, and the only other surface with this property is the punctured
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
(a torus minus an open disk).
Pants decompositions
The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
compact surface
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
with
boundary components to be
, and for a non-connected surface take the sum over all components. Then the only surfaces with negative Euler characteristic and complexity zero are
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
s of pairs of pants. Furthermore, for any surface
and any
simple closed curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
on
which is not
homotopic
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 ...
to a boundary component, the compact surface obtained by cutting
along
has a complexity that is strictly less than
. In this sense, pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic.
By a recursion argument, this implies that for any surface there is a system of simple closed curves which cut the surface into pairs of pants. This is called a ''pants decomposition'' for the surface, and the curves are called the ''cuffs'' of the decomposition. This decomposition is not unique, but by quantifying the argument one sees that all pants decompositions of a given surface have the same number of curves, which is exactly the complexity. For connected surfaces a pants decomposition has exactly
pants.
A collection of simple closed curves on a surface is a pants decomposition if and only if they are disjoint, no two of them are homotopic and none is homotopic to a boundary component, and the collection is maximal for these properties.
The pants complex
A given surface has infinitely many distinct pants decompositions (we understand two decompositions to be distinct when they are not homotopic). One way to try to understand the relations between all these decompositions is the ''pants complex associated to the surface''. This is a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
with vertex set the pants decompositions of
, and two vertices are joined if they are related by an elementary move, which is one of the two following operations:
*take a curve
in the decomposition in a one-holed torus and replace it by a curve in the torus intersecting it only once,
*take a curve
in the decomposition in a four-holed sphere and replace it by a curve in the sphere intersecting it only twice.
The pants complex is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
(meaning any two pants decompositions are related by a sequence of elementary moves) and has infinite
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
(meaning that there is no upper bound on the number of moves needed to get from one decomposition to the other). In the particular case when the surface has complexity 1, the pants complex is
isomorphic to the
Farey graph.
The
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of 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 ...
on the pants complex is of interest for studying this group. For example, Allen Hatcher and William Thurston have used it to give a proof of the fact that it is
finitely presented.
Pants in hyperbolic geometry
Moduli space of hyperbolic pants
The interesting hyperbolic structures on a pair of pants are easily classified.
: For all
there is a hyperbolic surface
which is homeomorphic to a pair of pants and whose boundary components are simple closed geodesics of lengths equal to
. Such a surface is uniquely determined by the
up to
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
.
By taking the length of a cuff to be equal to zero, one obtains a
complete metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
on the pair of pants minus the cuff, which is replaced by a
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
. This structure is of finite volume.
Pants and hexagons
The geometric proof of the classification in the previous paragraph is important for understanding the structure of hyperbolic pants. It proceeds as follows: Given a hyperbolic pair of pants with totally geodesic boundary, there exist three unique geodesic arcs that join the cuffs pairwise and that are perpendicular to them at their endpoints. These arcs are called the ''seams'' of the pants.
Cutting the pants along the seams, one gets two right-angled hyperbolic hexagons which have three alternate sides of matching lengths. The following lemma can be proven with elementary hyperbolic geometry.
: If two right-angled hyperbolic hexagons each have three alternate sides of matching lengths, then they are isometric to each other.
So we see that the pair of pants is the
double
A double is a look-alike or doppelgänger; one person or being that resembles another.
Double, The Double or Dubble may also refer to:
Film and television
* Double (filmmaking), someone who substitutes for the credited actor of a character
* ...
of a right-angled hexagon along alternate sides. Since the isometry class of the hexagon is also uniquely determined by the lengths of the remaining three alternate sides, the classification of pants follows from that of hexagons.
When a length of one cuff is zero one replaces the corresponding side in the right-angled hexagon by an ideal vertex.
Fenchel-Nielsen coordinates
A point in the Teichmüller space of a surface
is represented by a pair
where
is a complete hyperbolic surface and
a diffeomorphism.
If
has a pants decomposition by curves
then one can parametrise Teichmüller pairs by the Fenchel-Nielsen coordinates which are defined as follows. The ''cuff lengths''
are simply the lengths of the closed geodesics homotopic to the
.
The ''twist parameters''
are harder to define. They correspond to how much one turns when gluing two pairs of pants along
: this defines them modulo
. One can refine the definition (using either analytic continuation or geometric techniques) to obtain twist parameters valued in
(roughly, the point is that when one makes a full turn one changes the point in Teichmüller space by precomposing
with a
Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
Definition
Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ...
around
).
The pants complex and the Weil-Petersson metric
One can define a map from the pants complex to Teichmüller space, which takes a pants decomposition to an arbitrarily chosen point in the region where the cuff part of the Fenchel-Nielsen coordinates are bounded by a large enough constant. It is a
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
when Teichmüller space is endowed with the
Weil-Petersson metric, which has proven useful in the study of this metric.
Pairs of pants and Schottky groups
These structures correspond to
Schottky groups
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by .
Definition
Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we cal ...
on two generators (more precisely, if the quotient of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
by a Schottky group on two generators is homeomorphic to the interior of a pair of pants then its convex core is an hyperbolic pair of pants as described above, and all are obtained as such).
2-dimensional cobordisms
A cobordism between two ''n''-dimensional
closed manifolds is a compact (''n''+1)-dimensional manifold whose boundary is the disjoint union of the two manifolds. The
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of cobordisms of dimension ''n''+1 is the category with objects the closed manifolds of dimension ''n'', and
morphisms the cobordisms between them (note that the definition of a cobordism includes the identification of the boundary to the manifolds). Note that one of the manifolds can be empty; in particular a closed manifold of dimension ''n''+1 is viewed as an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
of the
empty set. One can also compose two cobordisms when the end of the first is equal to the start of the second. A n-dimensional topological quantum field theory (TQFT) is a monoidal functor from the category of ''n''-cobordisms to the category of complex vector space (where multiplication is given by the tensor product).
In particular, cobordisms between 1-dimensional manifolds (which are unions of circles) are compact surfaces whose boundary has been separated into two disjoint unions of circles. Two-dimensional TQFTs correspond to
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality th ...
s, where the circle (the only connected closed 1-manifold) maps to the underlying vector space of the algebra, while the pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative. Further, the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary, which completes the correspondence.
Notes
References
*
*
* {{cite book , last=Ratcliffe , first=John , title=Foundations of hyperbolic manifolds, Second edition , publisher=Springer , year=2006 , pages=xii+779 , isbn=978-0387-33197-3
Topology
Hyperbolic geometry
Geometry processing