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
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