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.
Previous Page Next Page