CHAPTER 2
The case
V3
=
M3
of Theorem I and Theorem II.
It will be convenient for us to start by presenting the proof of Theorem I in the
special case
V3
= M
3
; this proof, although not achieving full equivariance for the
action TTIM3
X
M
3
—• M
3
, will present special features which will come in very
handy when going to the equivariant representation Theorem II. In a subsequent
chapter we will also indicate how the proof of the special case of Theorem I can
be adapted for the case of the general open simply connected F
3
's; that is actually
easy.
Both the proofs of Theorems I and II will have a 3-dimensional part, producing
a 3-dimension representation (huge 3-dimensional object)—• (M
3
or
V3).
A big part of the our effort will go into this 3-dimensional representation. The
X2
will be gotten, afterwards, essentially as some sort of the 2-skeleton of the {huge
3-dimensional object } above. But that is an easier part of the story.
REMARKS.
A) There is another reason, of course, for paying special attention, even in the
case of Theorem I, to M
3
, since it is really the issue of
n^M3
which we are after
(see [PolO], [Poll],[Pol2]).
B) As already said, the proof of the special case of Theorem I still falls far
short of equivariance, in the specific
M3
case. But some weak form of TTIM3-
equivariance (to be made precise later on) will still be achieved, in our special
approach to Theorem I (with
V3
= M
3
). Suffices to say here that in the context
of the basic estimate (1.4) we also get now a UNIFORM BOUND. More precisely,
consider an arbitrary Riemannian metric on
M3
and lift it to M
3
(which gives a
Riemannian metric on
M3
which is well-defined up to a quasi-isometry.) For every
A 0 there exists a /i 0 so that for tight transversal £ of length inferior to A, we
have
card[lim(M2(/)nf)] \i.
But the "weak
TT\M3-equivariance
", established in the special case of Theorem
I, which is much more precise than the estimate above, plays a big role in the final
proof of Theorem II (which really is our main result in the present paper.)
So, for the time being, we will stick to a closed 3-manifold M
3
and to M
3
.
It is well-known that we can always represent
M3
as follows. We start with a
polyhedral 3-ball A with triangulated 9A, containing an even number of triangles
/ii, /i2, •. , ^2p- We are also given a fixed-point free involution
(2.1) 5 =
/
{ / i i , / i
2 )
. . . , / i
2 p
} - ^ 5
19
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