Taming Manifolds that are Wild at Infinity

Craig R. Guilbault

One of the best known theorems in the study of non-compact manifolds is found in L.C. Siebenmann's 1965 Ph.D. thesis. It gives necessary and sufficient conditions for the end of an open manifold to possess the simplest possible structure-that of an open collar.

Theorem 1 (Siebenmann)   A one ended open $n$-manifold $M^{n}$ ($n\geq6$) contains an open collar neighborhood of infinity iff each of the following is satisfied:

1. $M^{n}$ is inward tame at infinity,

2. $\pi_{1}\left( \varepsilon(M^{n})\right)$ is stable, and

3. $\sigma_{\infty}\left( M^{n}\right) \in\widetilde{K}_{0}\left( \mathbb{Z[}\pi_{1}(\varepsilon(M^{n}))]\right)$ is trivial.

One of the beauties of Siebenmann's theorem is the simple structure it places on the ends of certain manifolds. At the same time, this simplicity greatly limits the class of manifolds to which it applies. Indeed, many interesting and important non-compact manifolds are ``too complicated at infinity'' to be collarable. Frequently the condition these manifolds violate is $\pi_{1}$-stability. We will discuss an ongoing program to obtain variations on this theorem that apply to manifolds with non-stable fundamental groups at infinity.