1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 201

Proposition 4.19. The map Mo,n --t (S^2 )In of Proposition 4.18 is a homeomor-

phism onto its image.

Proposition 4.20. The image of the map Mo,n --t (S^2 )In of Proposition 4.18 is

a smooth submanifold of (S^2 )In.

Below we sketch the main ideas of the proofs. Full details are given in [20].

For any stable Riemann surface z , any a E T, and any i E {1, ... , n }, denote

(44) if ai =a,

if ai E Ta./3·

If a =f. ai, then one can think of Za.i as the unique singular point on the a-sphere,

through which it is connected to the sphere on which Zi lies, namely, the ai-sphere

(see Figure 15).

Figure 15. Marked points

Exercise 4.21. Let z = ({za.13}a.E13,{ai,zih~i9) be a stable Riemann surface

with n marked points. Prove that, if a, (3 E T with aE(J and Za.i =f. Za.f3, then
~=~. D

Proof of Proposition 4.18: The proof consists of four steps. The first two steps
are the definition of the map Mo,n --t (S^2 ln and their proofs are easy exercises.
The third step is injectivity. The fourth step identifies the image with Mn.

Step 1: Let z be a stable Riemann surface of genus zero with n marked points.
Then for any three distinct indices i, j, k E {1, ... , n} there exists a unique vertex

a ET such that Za.i =f. Za.j =f. Za.k =f. Za.i (see Figure 16).

Figure 16. A tree triangle

Step 2: Let z be a stable Riemann surface of genus zero with n marked points
and suppose that the integers i, j, k, l E {1, ... , n} are pairwise distinct. Then there
exists an a E T such that ( Za.i, Za.j, Za.k, Za.e) i ~ 3. The number
Wijke(z) := w(za.i, Za.j, Za.k, Za.e)