1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 193

to at least three other curves in the tree (at distinct points). This condition ensures

that there are only finitely many automorphisms which preserve the curve.

4.2. Stable maps

The concept of a stable map was introduced by Kontsevich [25]. We begin with
the definition of a tree as a connected graph without cycles. Think of a tree as a
finite set (of vertices) equipped with a relation E such that two vertices are related
by E iff they are connected by an edge.

Definition 4.2. A tree is a finite set T with a relation E C T x T satisfying the

following axioms.

(symmetric) If aE(3 then (3Ea.

(anti-reflexive) If aE(3 then a =I-(3.

(connected) For all a, (3 E T with a =I-(3 there exist 10, ... , Im E T with

10 =a and Im = (3 such that liEli+l for all i.

(no cycles) If 10, ... , Im E T with liEli+l and Ii =I-li+2 for all i then

lo =I-Im·

A map f: (T, E)---> (T, E) is called a tree homomorphism if f-^1 (&.) is a tree for

all a ET and, for all a,(3 ET with aE(3 and f(a) =I-f((3), we have f(a)Ef((3).

It is called a tree isomorphism if it is bijective and both f and f-^1 are tree

homomorphisms.

I I

I '

I

I

I

a

'

Figure 13. Trees

Figure 14. Stable maps

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