Pair of pants (mathematics)

The pair of pants is the compact genus-zero surface with boundary having three boundary components. It is obtained from the sphere by deleting the interiors of three disjoint discs and is homotopy equivalent to a wedge of two circles.

A Y-piece is a compact curvature-$-1$ hyperbolic surface homeomorphic to a pair of pants, with three geodesic boundary curves. Its labelled boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3>0$ determine it up to isometry. Cutting along the three mutually perpendicular seams produces two congruent right-angled hyperbolic hexagons with alternating sides ${\ell }_i/2$. Conversely, doubling the hexagon along its other three sides constructs the Y-piece for arbitrary positive lengths.

An X-piece is a curvature-$-1$ four-holed sphere with geodesic boundary, formed by gluing two Y-pieces along equal-length cuffs. For a specified separating cuff and labelled outer boundaries, four outer lengths, the internal cuff length $s>0$, and a twist $\tau$ determine the metric. The gluing is orientation-reversing on boundary circles, with relative arclength displacement $\tau$ measured from chosen seam endpoints. Without markings, the gluing displacement is taken modulo $s$; with a marking, $\tau \in \mathbb{R}$ records Dehn twists as well. Changing the chosen separating curve can give different parameter descriptions of the same unmarked surface.

Orient the three boundary classes $c_1,c_2,c_3$ so that $c_1+c_2+c_3=0$ in $H_1(N;\mathbb{Z})$. A homeomorphism with orientation sign $ϵ$ and boundary permutation $\sigma$ sends $c_i$ to $ϵc_{\sigma (i)}$. Its trace on the rank-two quotient of ${\mathbb{Z}}^3$ by $c_1+c_2+c_3$ is therefore $ϵ(\#\mathrm{Fix}\sigma -1)$.