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