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- Counting the number of ways to obtain a surface by
gluing the sides of a polygon, preprint (1992).
This was an exercise, but not an easy one.
Harer and Zagier [HZ] proved that the virtual Euler
characteristic of the mapping class group of the
orientable closed surface of genus g is
A crucial part of their proof is the recurrence formula
is the number of ways to obtain an orientable
surface of genus g by pairing the sides of a 2n-gon.
Their proof is long and uses heavy machinery; they ask for an
elementary one. My paper gives an elementary proof,
though not very short.
- J. Harer and D. Zagier, The Euler characteristic of
the moduli space of curves, Invent. Math. 85 (1986),