Pair of pants

Pair of pants

In mathematics, a pair of pants is a simple two-dimensional surface resembling a pair of pants. 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. 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.

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 points removed. 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).


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