# Equivariant, Almost-Arborescent Representations of Open Simply-Connected 3-Manifolds; A Finiteness Result

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*V. Poénaru; C. Tanasi*

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

#### Table of Contents

# Table of Contents

## Equivariant, Almost-Arborescent Representations of Open Simply-Connected 3-Manifolds; A Finiteness Result

- Contents v6 free
- Chapter 1. Introduction 18 free
- Acknowledgments 1825 free

- Chapter 2. The case V[sup(3)] = M[sup(3)] of Theorem I and Theorem II. 1926
- A singular handlebody attached to T[sup(F)]→M[sup(3)] 2128
- Collared spaces and doubly collared handles 2330
- An abstract symmetry-forcing procedure 3037
- A locally finite train-track manifold attached to T[sup(F)]→M[sup(3)] 3239
- Construction of a π[sub(i)]M[sup(3)]-equivariant Θ(β)⊃Θ(α) 4249
- Partial Foliations 5663

- Chapter 3. The accumulation pattern of the double point M[sub(2)](ƒ) ⊃ X[sup(2)] 7582
- Chapter 4. Arbitrary open simply-connected 3-manifold 8592
- Bibliography 8794