the criteria for 3-dim manifolds diffeomorphic to $\mathbb{R}^3$

Question

In Schoen and Yau's paper "Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature", they mentioned “If $M^3$ is contractible, to prove it is diffeomorphic to $\mathbb{R}^3$, from a topological result by Stallings "Group Theory and Three DimensionalManifolds", it suffices to prove that $M^3$ is simply connected at infinity and irreducible.”

I checked Stallings’ book, did not find the exact result they referred to. Can anyone tell me which theorem in Stallings’ book they referred to or any other place to find the detailed proof of the above statement? Thanks.

Answer 1

I am not sure what Schoen and Yau had in mind. Any open contractible $n$-manifold that is simply-connected at infinity is homeomorphic to $\mathbb R^n$. This is due to Stallings [1] if $n\ge 5$ and to Guilbault [2] if $n=4$, while the case $n=3$ follows from the result of Husch and Price [3] and the non-existence of fake $3$-cells by Perelman's solution of the Poincaré conjecture.

[1] J.~Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481--488

[2] C.~R. Guilbault, An open collar theorem for $4$-manifolds, Trans. Amer. Math. Soc. 331 (1992), no.~1, 227--245.

[3] L.~S. Husch and T.~M. Price, Finding a boundary for a $3$-manifold, Ann. of Math. 91 (1970), 223--235.

Since the question is about dimension $3$ let me give more details. By Perelman there is no fake $3$-cells. Now Theorem 2 of [3] implies that any contractible $3$-manifold that is simply-connected at infinity is homeomorphic to the interior of a compact $3$-manifold $N$ (in fact the result in [3] gives a general criterion of when one can attach a boundary to a $3$-manifold). Now $N$ is a contractible $3$-manifold whose boundary is a $2$-sphere, and again invoking Perelman we conclude that $N$ is a $3$-disk, up to homeomorphism. Finally, by Moise homeomorphic $3$-manifolds are diffeomorphic.

Answer 2

Husch and Price is the right reference, but I thought I could give a quick sketch of a proof.

Let $X^3$ be simply connected at infinity and contractible. For every compact $C \subset X$, there exists a compact $D$ containing $C$ such that $π_1(X-D)\to π_1(X-C)$ is trivial. Taking a regular neighborhood, we may assume that $D$ is a compact submanifold with boundary. Compress $∂D$ as much as possible in the complement of $C$ to obtain a compact incompressible surface $S \subset X-C$. If $S$ is not a union of 2-spheres, then each non-trivial component is $π_1$-injective in $X-C$ by the loop theorem. But the fundamental group of this surface is in the image of $π_1(∂D) \to π_1(X-C)$, which is trivial, a contradiction. Thus, $S$ must consist of a collection of 2-spheres. Since $X$ is contractible, each such sphere bounds a contractible submanifold, which is a 3-ball by the Poincaré conjecture (and I suppose van Kampen's theorem). Hence $C$ is contained in a 3-ball. Thus, $X$ is a nested union of 3-balls, and is therefore homeomorphic to $R^3$.