A SINGULAR HANDLEBODY ATTACHED TO T A M6 21

We also have an obvious commutative diagram which connects this "left action"

°f Grea on T, to the standard left action of

7r\M3

on M

3

_ 9 _

T ^ T

(2.5) F = F ( A o

F(A0)

M 3 ^ M 3

REMARK.

The tree-like union of fundamental domains T is clearly not locally-

finite. Also it is singular along its 1-skeleton; there T is not a 3-manifold and F is

not immersive.

A singular handlebody attached to T — M

3

Starting with (2.3), we have a canonical decomposition of the universal covering

space into fundamental domains

(2.6) M

3

=

(J g&,

geniM3

which is such that T — M

3

is "simplicially" non-degenerate; our F projects

the tree-like decomposition T = J2n(^r ~9^ o n t o (2.6). The (2.6) is a cell-

__

g^

red

decomposition of M 3 , which is simplicial, as far as its 2-skeleton is concerned.

There is an obvious canonical way to change (2.6) into a 7TiM3-equivariant handle-

body decomposition

(

oo \ / oo \ / oo

E«?)U(E«J)U(E«?

with the following features.

i) There is a canonical bijection

(2.7.1) {A-cells of (2.6), call them ax} ^ +

^-^ {handles of index A, call them

sA,

in (2.7)}.

ii) The various sf,s5 (with the same index e ) are pairwise disjoint and the

handle, of index A, sx = Bx x B3~x is glued to 0(U2=o(££i 4)) c ^

along its attaching zone dBx x B3~x.

iii) It will be convenient to use the following terminology for an n-dimensional

handle, of index A,

Bx

x

Bn~x.

We will call

Bx

«

Bx

x (center of

Bn~x)

the

core,

Bn~x

w (center of £

A

) x £

n

~

A

the co-core,

dBx

x 5

n

~

A

= {9(core)

x (co-core)} the attaching zone and

Bx

x

dBn~x

— {(core xd(co-core)}

the lateral surface. With this, if A /x, then {attaching zone rf} fl {lateral

surface sx} ^ 0 if and only if cr^ D ax ^ 0. In other words, except for the

fact that distinct handles of the same index are disjointed (see ii) above),

the incidence relations among the handles cr|'s are exactly like those among

the simplexes cr|'s.