8 YAEL KARSHON

FIGURE

3. Delzant polygons for Example 2.8

The pre-image in the manifold of a vertex of the polygon is a fixed

point for the torus action. The pre-image of an edge with slope k/b

is an invariant 2-sphere whose stabilizer is the subgroup of S1 x S*1

consisting of the elements of the form (Afc, A-6). The pre-image of the

interior of the polygon consists of free torus orbits. These facts follow

from the local normal form for Hamiltonian torus actions and from the

connectedness of the level sets of the moment map, and are explained

in Delzant's paper [De].

If we now restrict the action to the sub-circle {e} x 5

1

, we get a

compact four dimensional Hamiltonian S^-space. The moment map

for the Sfl-action is the T-moment map composed with the projec-

tion R2 — M to the second coordinate. The fixed surfaces are the

pre-images, under the T-moment map, of the horizontal edges of the

Delzant polygon. Such a surface has genus zero, and its normalized

symplectic area is equal to the length of the corresponding horizontal

edge. The isolated fixed points are the pre-images of those vertices of

the polygon that do not lie on horizontal edges. The Z^-spheres are

the pre-images of edges with slope ±k/b in reduced form, where b is

relatively prime to k. With this information, it is easy to construct the

graph for the

S1

space out of the Delzant polygon.

Remark 2.7. In our figures of Delzant polygons, the dots mark the

weight lattice

Z2

in

R2.

Example 2.8. The Delzant spaces whose polygons are drawn in figure

3 give Hamiltonian

51-spaces

with the same graph, which is the graph

drawn in Figure 2. These

S1

-spaces are all isomorphic to the one

described in Example 2.4. The three polygons all correspond to S2 x S2,

with its standard torus action, composed with an automorphism of the

torus that preserves the second sub-circle.