Manifolds with non-stable fundamental groups at infinity, III

This is the third in a series of papers aimed at generalizing Siebenmann’s famous PhD thesis [13] so that the results apply to manifolds with nonstable fundamental groups at infinity. Siebenmann’s work provides necessary and sufficient conditions for an open manifold of dimension ≥ 6 to contain an open collar neighborhood of infinity, ie, a manifold neighborhood of infinity N such that N ≈ ∂N × [0, 1). Clearly, a stable fundamental group at infinity is necessary in order for such a neighborhood to exist. Hence, our first task was to identify a useful, but less rigid, ‘end structure’ to aim for. We define a manifold Nn with compact boundary to be a homotopy collar provided ∂Nn ↪→ Nn is a homotopy equivalence. Then define a pseudo-collar to be a homotopy collar which contains arbitrarily small homotopy collar neighborhoods of infinity. An open manifold (or more generally, a manifold with compact boundary) is pseudo-collarable if it contains a pseudo-collar neighborhood of infinity. Obviously, an open collar is a special case of a pseudo-collar. Guilbault [7] contains a detailed discussion of pseudo-collars, including motivation for the definition and a variety of examples—both pseudo-collarable and non-pseudo-collarable. In addition, a set of three conditions (see below) necessary for pseudo-collarability—each analogous to a condition from Siebenmann’s original theorem—was identified there. A primary goal became establishment of the sufficiency of these conditions. At the time [7] was written, we were only partly successful at attaining that goal. We obtained an existence theorem for pseudo-collars, but only by making an additional assumption regarding the second homotopy group at infinity. In this paper we eliminate that hypothesis; thereby obtaining the following complete characterization.