Pants (math)

A pair of pants is bordered by 3 closed curves.

In topology , a branch of mathematics , are areas of gender 0 3 edge components as trousers (ger .: pair of pants called). Its interior is homeomorphic into a triple-dotted sphere. Most topological surfaces can be broken down into trousers ("trouser decomposition").

construction

A pant can be obtained from a two-dimensional sphere by cutting out three discs.

Alternatively, you can get pants made of two hexagons by gluing the corresponding pairs of red sides (picture below right) together. The pairs of blue sides then correspond to the three edge components.

Trouser dismantling

The area of genus 2 with 4 edge components can be divided into 6 pants.

You can glue multiple pants along some of their edge components, creating more intricate surfaces. The corresponding decomposition of the resulting surface is known as the trouser decomposition .

A surface of the gender with edge components has a trouser decomposition if and only if ${\ displaystyle g}$ ${\ displaystyle n}$

{\ displaystyle 2 (g-1) + n> 0}

,

so if either or or is. In general, a surface can have several different trouser cuts. The number of pants in each pants division is . The number of decomposing curves is . ${\ displaystyle g \ geq 2}$ ${\ displaystyle g = 1, n \ geq 1}$ ${\ displaystyle g = 0, n \ geq 3}$ ${\ displaystyle 2 (g-1) + n}$ ${\ displaystyle 3 (g-1) + n}$

Hyperbolic geometry

Pants are made up of two such hexagons.

For every triple of positive real numbers there is a hyperbolic metric on the pants, so that the three edge components are closed geodesics of the lengths . (This follows from the fact that there is a right-angled hyperbolic hexagon that is unambiguous except for congruence with the lengths of the edges drawn in blue on the right, as well as from the division of a pair of pants into two hexagons.) ${\ displaystyle (\ gamma _ {1}, \ gamma _ {2}, \ gamma _ {3})}$ ${\ displaystyle \ gamma _ {1}, \ gamma _ {2}, \ gamma _ {3}}$ ${\ displaystyle \ gamma _ {1} / 2, \ gamma _ {2} / 2, \ gamma _ {3} / 2}$

The trouser decomposition can be used to define the Fennel-Nielsen coordinates on the Teichmüller space: For a closed area of the gender , a trouser decomposition with decomposing curves is fixed . The lengths of these curves together with the twist parameters assigned to the curves define parameters for the Teichmüller space of the hyperbolic metrics on the surface. ${\ displaystyle g}$ ${\ displaystyle 3 (g-1)}$ ${\ displaystyle 6 (g-1)}$

(1 + 1) -dimensional topological quantum field theories

A (n + 1) -dimensional topological quantum field theory assigns contiguous, self-n-dimensional manifolds each having a vector space (and their disjoint associations, the tensor product of each vector spaces) as well each bordism a vector spaces of the corresponding edges vector spaces to, certain axioms must be met .

A pair of pants can (depending on the arrangement) be viewed as a border between and or as a border between and . In a (1 + 1) -dimensional topological field theory, a pair of pants defines a multiplication in the first case, and a co-multiplication in the second. ${\ displaystyle S ^ {1} \ sqcup S ^ {1}}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle S ^ {1} \ sqcup S ^ {1}}$

One can show that a (1 + 1) -dimensional field theory with this multiplication and multiplication defines a Frobenius algebra .

literature

Albert Fathi, François Laudenbach, Valentin Poénaru: Thurston's work on surfaces. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. (= Mathematical Notes. 48). Princeton University Press, Princeton, NJ 2012, ISBN 978-0-691-14735-2 .
Riccardo Benedetti, Carlo Petronio: Lectures on hyperbolic geometry. University text. Springer-Verlag, Berlin 1992, ISBN 3-540-55534-X .
Joachim Kock: Frobenius algebras and 2D topological quantum field theories. (= London Mathematical Society Student Texts. 59). Cambridge University Press, Cambridge 2004, ISBN 0-521-83267-5 .

Individual evidence

↑ Proposition B.2.5. in Benedetti-Petronio (op.cit.)
↑ see Fathi-Laudenbach-Poénaru (op.cit.)
↑ Theorem B.4.17. in Benedetti-Petronio (op.cit.)
↑ see Kock (op.cit.)

[1] Proposition B.2.5. in Benedetti-Petronio (op.cit.)

[2] see Fathi-Laudenbach-Poénaru (op.cit.)

[3] Theorem B.4.17. in Benedetti-Petronio (op.cit.)

[4] see Kock (op.cit.)