Pair of pants (mathematics)

This article is about an object in hyperbolic geometry. For the article of clothing, see trousers.

A pair of pants

Six pairs of pants sewn together to form an open surface of genus two with four boundary components.

In mathematics, a pair of pants is a simple two-dimensional surface resembling a pair of pants: topologically, it is a sphere with 3 holes in it. Pair of pants admit hyperbolic metrics, and their isometry class is determined by the lengths of the boundary curves (the cuff lengths), or dually the distances between the boundaries (the seam lengths).

In hyperbolic geometry all three holes are considered equivalent – no distinction is made between "legs" and "waist". In cobordism theory the holes are not equivalent – a pair of pants is a cobordism between one circle (the "waist") and two circles (the "legs").

Hyperbolic geometry

In hyperbolic geometry, pairs of pants are sewn together, leg to leg, or leg to waist (there is no distinction between the legs and the waist), to create Riemann surfaces of arbitrary genus; conversely, Riemann surfaces can be cut into pairs of pants by cutting along closed geodesics. Because the "legs" can be twisted before being sewn together, there is a large amount of freedom in how the pants can be assembled. This ambiguity is then the moduli space of the Riemann surface, which has complex dimension $3(g-1)=3g-3$ for $g>1.$

Formally, a pair of pants consists of two hexagonal fundamental polygons stitched together at every other side. Topologically, a pair of pants is the two-sphere $S^{2}$ with three open disks removed, or equivalently the disk with two open disks removed. This is a deformation retract of the thrice-punctured sphere (a sphere with three points removed), though these are not homeomorphic – the thrice-punctured spheres is not compact, and has no boundary components. Geometrically, a thrice-punctured sphere corresponds to pants where the cuff length is zero – where instead of a boundary circle, one has a cusp – compare ideal triangle.

A pair of pants, as a (subset of a) thrice-punctured sphere, admits a hyperbolic structure, unlike the unpunctured or once or twice punctured spheres (sphere, plane, annulus), which admit positive curvature, zero curvature, and zero curvature, respectively – compare Little Picard theorem.

It is homotopy equivalent to the wedge sum of two circles, and thus has fundamental group isomorphic to the free group on two generators (one generator for each circle).

A pair of pants is analogous to a fattened up hyperbolic triangle, and is frequently so drawn schematically, with the seams as the sides and cuffs at vertices. Compare SSS and AAA congruence of hyperbolic triangles to pairs of pants being determined by seam length or cuff lengths.

Cobordism theory

This link cobordism between the Hopf link and the unlink is topologically a pair of pants.

In cobordism theory, a pair of pants is a cobordism between a single circle and two circles (the waist and the legs), and, together with the fact that all compact connected 1-manifolds are circles, shows that the cobordism group of 1-manifolds is trivial. This also follows because a circle bounds a disk, and from this point of view a pair of pants is the connected sum of a cylinder (identity cobordism of a circle) and a disk (null-cobordism of a circle).

Topological quantum field theory

Further information: Topological quantum field theory

In topological quantum field theory (TQFT), a pair of pants corresponds to multiplication or comultiplication in a Frobenius algebra, depending on which side is up, as follows.

An $(n+1)$-dimensional TQFT, in Atiyah's axiomatization, is a symmetric monoidal functor from the category of $(n+1)$-dimensional cobordism between n-dimensional manifolds to the category of vector spaces. In other words, it takes n-dimensional manifolds to vector spaces, disjoint unions of manifolds to tensor products of vector spaces, and cobordisms between manifolds to maps between vector spaces, satisfying suitable axioms. (1+1)-dimensional TQFTs correspond to Frobenius algebras, 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.

References

• The Geometry and Topology of Three-Manifolds, William P. Thurston