CHAPTER 1

Introduction

We will start by describing in very broad lines what this paper, which replaces

and supersedes the older preprint [Pol3], by the first author, is all about; precise

definitions and statements will follow later on in the present chapter. The papers

[Ga], [Po3] (and the easy initial pages of [Po4]), contain all the basic material

necessary for the present work. In [Po3] the first author has given a certain way

of presenting homotopy 3-spheres

E3,

the collapsible pseudo-spine represen-

tation theorem; what the theorem in question says is that given

E3

we can find

£

a collapsible finite 2-complex K2 and a non-degenerate simplicial map K2 — E3

with the following three properties

i) The map / is a generic immersion, except at a finite set of points Sing(/) C

K2 which are the undrawable singularities from figure 1.1. We sometimes

also call them admissible singularities.

ii) The double points of / are commanded by the singularities x G Sing(/), i.e.

they can be "zipped"; in slightly more precise terms this means that the

smallest equivalence relation \£(/) C K2 x K2, compatible with / , which

is such that K2/^(f) —• E3 is an immersion (i.e. a non-singular map),

is actually the equivalence relation $(/ ) C

K2

x

K2

defined by the map /

itself (i.e. (x,y) G $(/ ) iff fx = fy.) These equivalence relations \£, £ are

discussed, in detail, in [Pol].

iii) The open regular neighbourhood

Nbd(/i^2)

C

E3

of

fK2

C

E3

can be

isotoped, via a non-ambient isotopy, to

E3-{a

finite set of points}.

In [Po3] the first author has also introduced the concept of an almost-collapsi-

ble space; this is a finite complex obtained by adding to a collapsible 2-complex

Q a finite number of 2-cells, glued to Q along their boundaries. One can define al-

most-collapsible pseudo-spine representations for

E3

and a very easy remark

(see [Po3]) tells us that from an almost-collapsible pseudo-spine representation for

E3

we can immediately get a collapsible one; the converse is obvious, of course. So

we might as well think from now on that the K2 appearing in our representation

K2

—

E3

(with the properties i), ii), iii)) is almost collapsible, rather than

just collapsible. It is this equivalent formulation, of the collapsible pseudo-spine

representation theorem [Po3], that will be extended, in the present work, to the

case of open simply-connected 3-manifolds. So, in the present paper we consider

an open simply connected 3-manifolds V3 in lieu of E3 and an infinite but locally

finite 2-dimensional simplicial complex X2 in lieu of K2; this X2 will be almost

arborescent (this notion, which we will define precisely, later on, is the substitute

for almost-collapsibility in the context of infinite complexes).

1