When one extends the (almost) collapsible pseudo-spine
representation theorem for homotopy $3$-spheres [**Po3**] to open
simply connected $3$-manifolds $V^3$, new phenomena appear:
at the source of the representation, the set of double points is, generally
speaking, no longer closed. We show that at the cost of replacing
$V^3$ by $V_h^3 = \{ V^3 \text{ with very many holes}\}$,
we can always find representations $X^2
\stackrel{f}{\rightarrow} V^3$ with $X^2$ locally finite and
almost-arborescent, with $\Psi (f)=\Phi (f)$, with the open regular
neighbourhood (the only one which is well-defined here)
Nbd$(fX^2)=V^3_h$ and such that on any precompact tight transversal to
the set of double lines, we have only finitely many limit points (of
the set of double points). Moreover, if $V^3$ is the universal
covering space of a closed $3$-manifold, $V^3=\widetilde
M^3$, then we can find an $X^2$ with a free $\pi_1M^3$
action and having the **equivariance** property $f(gx)=gf(x)$,
$g\in \pi_1M^3$. Having simultaneously all these properties for
$X^2\stackrel{f}{\rightarrow} \widetilde M^3$ is one of the steps in
the first author's program for proving that $\pi_1^\infty \widetilde
M^3=~0$, [**Po11**, **Po12**]. Achieving equivariance is far from
being straightforward, since $X^2$ is gotten starting from a tree of
fundamental domains on which $\pi_1M^3$ cannot, generally speaking,
act freely. So, in this paper we have both a representation theorem for general
($\pi_1=0$) $V^3$'s and a harder **equivariant**
representation theorem for $\widetilde M^3$ (with $gfX^2=fX^2, \,
g\in\pi_1M^3$), the proof of which is **not** a
specialization of the first, “easier” result. But,
finiteness is achieved in both contexts. In a certain sense, this
finiteness is a best possible result, since if the set of limit points
in question is $\emptyset$ (i.e. if the set of double points is
closed), then $\pi_1^\infty V_h^3$ (which is always equal to
$\pi_1^\infty V^3$ ) is zero. In [**PoTa2**] it was also shown
that when we insist on representing $V^3$ itself, rather than
$V_h^3$, and if $V^3$ is **wild**
($\pi_1^\infty\not =0$), then the transversal structure of the set of
double lines can exhibit chaotic dynamical behavior. Our finiteness
theorem avoids chaos at the cost of a lot of redundancy (the same double point
$(x, y)$ can be reached in many distinct ways starting from the
singularities).

Readership

Graduate students and research mathematicians interested in
manifolds, cell complexes, dynamical systems, and ergodic theory.