Equivariant, Almost-Arborescent Representations of Open simply-Connected 3-Manifolds; A Finiteness result

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Abstract

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 V3 by Vh3 = {V3 with very many holes }, we can always find representations X2 →f V3 with X2 locally finite and almost-arborescent, with Ψ (f) = Φ(f), with the open regular neighbourhood (the only one which is well-defined here) Nbd(f X2) = Vh3 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 V3 is the universal covering space of a closed 3-manifold, V3 = M̃3, then we can find an X2 with a free π1M3 action and having the equivariance property f(gx) = gf(x), g ∈ π1M3. Having simultaneously all these properties for X2→f M̃3 is one of the steps in the first author's program for proving that π1∞M̃3 = 0, [Poll],[Pol2]. Achieving equivariance is far from being straightforward, since X2 is gotten starting from a tree of fundamental domains on which π1M3 cannot, generally speaking, act freely. So, in this paper we have both a representation theorem for general (π1 = 0) V3's and a harder equivariant representation theorem for Mπ3 (with gf X2 = fX2, g ∈ π1 M3), 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 ∅ (i.e. if the set of double points is closed), then π1∞Vh 3 (which is always equal to π1∞V 3 ) is zero. In [PoTa2] it was also shown that when we insist on representing V3 itself, rather than Vh33, and if V3 is wild (π1∞ ≠ 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).
Lingua originaleEnglish
pagine (da-a)0-89
Numero di pagine90
RivistaMemoirs of the American Mathematical Society
Volume169
Stato di pubblicazionePublished - 2004

All Science Journal Classification (ASJC) codes

  • ???subjectarea.asjc.2600.2600???
  • Applied Mathematics

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