Hint to Problem 5

Suppose you have a Frobenius algebra. Show that if you assign the
product V \tensor V --> V to the pair of pants like this >-, the
linear dual (taking into account the identification of V with V^* via
the scalar product on V) V --> V \tensor V to the pair of pants like
this -< (the difference with the first pair of pants is in the numbers
of inputs and outputs), and the unit or the trace to a cap in T(0,1)
or T(1,0), respectively, then algebraic properties of a Frobenius
algebra allow you to identify operators corresponding to sewn pairs of
pants like that

 \            \                                            \/
 /\__   ~      \__ , the reverse of that, and   >---<  ~    \
  /           \/                                             \
 /            /                                              /\

(all directions on edges assumed to go from left to right), as well as
to identify the operators corresponding to a pair of pants (directed
either way) with a leg capped off with the identity operator.

Then see that any oriented surface with a few (>=0) in-holes and a few
(>=0) out- holes may be cut into pairs of pants with designated inputs
and outputs and caps. Show that you may go through the above
transformations from any such decomposition into pairs of pants to a
standard one, determined only by the numbers of inputs, outputs and
the genus, such as this:

 \    /\       
 /\__/  \__/
  /  \  /  \   
 /    \/