18

1. INTRODUCTION

representation (1.3) is HIGHLY

REDUNDANT

in the sense that any given double point

(x,y)eM2(f)cX2xX2

of the map / can be reached in many ways, via zippings, starting with the singu-

larities of / . (Notice our notations for double points: M2(f) C X

2

, but

M2{f)

C

X2

x X

2

, with an obvious projection on the first factor

M2(f)

— M2(/)). Very

much linked with this redundancy is the fact that in order to get our almost ar-

borescent representation, we had to go from V3 to V^f, thereby highly increasing the

second homotopy group 7T2- In the same spirit, when we realize / by elementary 0(i)-

moves, many of these will be 0(3)'s and each of these creates a new ^-generator.

In our paper [PoTa2], it is explained that such a high redundancy is, very likely,

unavoidable for a wild 3-manifold

V3

(with

7rf°V3

=£ 0 like the Whitehead man-

ifold), if one insists in getting an (almost) arborescent representation satisfying

the SECOND FINITENESS property (1.4) from our theorem above. More explicitly

it is shown in [PoTa2] that for the most natural arborescent representation of the

Whitehead manifold

Wh3,

with Nbd(/X

2

) =

Wh3

(rather than

{Wh3

with very

many holes}), the well-known Julia sets from the dynamics of quadratic polynomi-

als occur quite naturally as lim(A D M2(/)), for appropriate tight transversals.

In other words the FINITENESS in our theorem is achieved at the price of a lot of

REDUNDANCY.

ACKNOWLEDGMENTS

The first author wants to thank Danny Calegari, David Gabai, Frangois Lau-

denbach, Barry Mazur and Panos Papasoglu for very helpful conversations con-

cerning the present paper. In particular, both David and Barry, independently

of each other, have made useful cautionary remarks concerning the issue of

ix\M3-

equivariance in our main theorem; very special thanks are actually due to David who

followed closely the development of the present work, during the years, helped us

straighten out many tricky issues, like for instance the issue of frustration (=non-

zero holonomy), and who offered innumerable useful comments, encouragements

and advice, during quite a long span of time.