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Agelos
Agelos
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Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.

Question 1: If $M$ is 1-ended, must it be homeomorphic with $\mathbb{R}^3$?

More generally:

Question 2: Is $M$ determined up to homeomorphism by its number of ends?

(By a classical result of Hopf, this number is 1, 2, or uncountably infinite, I think.)

More generally, I'm interested in results of the form: if a 3-manifold $M$ covers a compact manifold/orbifold, then it cannot be as wild as a generic open 3-manifold such as e.g. the Whitehead manifold.

Edit: Having discussed this with an expert, I believe that my questions above boil down to the following conjecture:

Conjecture: Let $G$ be the fundamental group of a closed 3-manifold $M$. Then $G$ is simply connected at infinity. (Equivalently, the universal cover of $M$ is simply connected at infinity.)

This conjecture is implicit in this paper by Funar & Otera: https://www-fourier.ujf-grenoble.fr/~funar/funote.pdf

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.

Question 1: If $M$ is 1-ended, must it be homeomorphic with $\mathbb{R}^3$?

More generally:

Question 2: Is $M$ determined up to homeomorphism by its number of ends?

(By a classical result of Hopf, this number is 1, 2, or uncountably infinite, I think.)

More generally, I'm interested in results of the form: if a 3-manifold $M$ covers a compact manifold/orbifold, then it cannot be as wild as a generic open 3-manifold such as e.g. the Whitehead manifold.

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.

Question 1: If $M$ is 1-ended, must it be homeomorphic with $\mathbb{R}^3$?

More generally:

Question 2: Is $M$ determined up to homeomorphism by its number of ends?

(By a classical result of Hopf, this number is 1, 2, or uncountably infinite, I think.)

More generally, I'm interested in results of the form: if a 3-manifold $M$ covers a compact manifold/orbifold, then it cannot be as wild as a generic open 3-manifold such as e.g. the Whitehead manifold.

Edit: Having discussed this with an expert, I believe that my questions above boil down to the following conjecture:

Conjecture: Let $G$ be the fundamental group of a closed 3-manifold $M$. Then $G$ is simply connected at infinity. (Equivalently, the universal cover of $M$ is simply connected at infinity.)

This conjecture is implicit in this paper by Funar & Otera: https://www-fourier.ujf-grenoble.fr/~funar/funote.pdf

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Agelos
Agelos
  • 1.9k
  • 10
  • 17

How wild can an open topological 3-manifold be if it has a compact quotient?

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group.

Question 1: If $M$ is 1-ended, must it be homeomorphic with $\mathbb{R}^3$?

More generally:

Question 2: Is $M$ determined up to homeomorphism by its number of ends?

(By a classical result of Hopf, this number is 1, 2, or uncountably infinite, I think.)

More generally, I'm interested in results of the form: if a 3-manifold $M$ covers a compact manifold/orbifold, then it cannot be as wild as a generic open 3-manifold such as e.g. the Whitehead manifold.