New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

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
 SEARCH THIS BOOK:
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

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).

• The case $$V^3=\widetilde M^3$$ of Theorem I and Theorem II
• The accumulation pattern of the double point $$M_2(f)\subset X^2$$