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Equivariant, Almost-Arborescent Representations of Open Simply-Connected 3-Manifolds; A Finiteness Result
V. Poénaru, Université Paris Sud, Orsay, France, and C. Tanasi, Palermo, Italy
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Memoirs of the American Mathematical Society
2004; 89 pp; softcover
Volume: 169
ISBN-10: 0-8218-3460-6
ISBN-13: 978-0-8218-3460-2
List Price: US$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/169/800
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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.

Table of Contents

  • Introduction
  • The case \(V^3=\widetilde M^3\) of Theorem I and Theorem II
  • The accumulation pattern of the double point \(M_2(f)\subset X^2\)
  • Arbitrary open simply-connected 3-manifold
  • Bibliography
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