## Surveys in Applied Mathematics. Essays Dedicated to S.M. by N. Metropolis, S. Orszag, G.-C. Rota

By N. Metropolis, S. Orszag, G.-C. Rota

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For instance, for the Gauss Sphere (Fig. 3) we could take the triangulation with OL2 = 6, οίχ = 9, and oc0 = 5. The number e = oc0 — oc1 + oc2 30 RAOUL BOTT 00 FIGURE 3 is then called the Euler number of the triangulation, and, as was already essentially known to Euler, the following theorem is valid: THEOREM 2. The Euler number of a triangulation of a surface X is independent of the triangulation. Thus e{X) is well defined, and this invariant of X is related to the genus of X by the formula e(X) = 2-2g(x).

The number δ = δ( p, q) of linearly independent regular ^-differentials is computed by the RiemannRoch theorem: a _ \P if 0 = 1. Ι(2ϊ-ΐΧ*-1) ^ ί>1. { r s n ; In other words, δ does not depend on the presence or the number of nodes. ,Fd be linearly independent regular ^-differentials on X. They determine a so called q-canonical holomorphic mapping of X into the (complex) projective space Ρδ _x . ,Λ(0)}. 52 LIPMAN BERS If P is a node, the image of P has homogeneous coordinates («1 > «2 i —> α δ) where ^· is the residue of Fj at a puncture (on a part of X) corresponding to P.

In the present paper we are concerned primarily with certain boundary points of S p belonging to δ(3 ρ . ,/>}, | / | = number of elements in /. ~3 for which (i) the gj(r, ·), j Φ / , are defined and are the free generators of a Schottky group (which we still denote by GT), (ii) the ^(r) with tel vanish, (iii) the 2\I \ points ^ 2i _ 1 (r), Λ2;·(Τ), j e / , are distinct, and (iv) lie in a suitably chosen standard fundamental region for GT belonging to ^ ( r , ·), j\$I. (A fundamental region with the above property will be still called a τ-region.