ENDS OF MANIFOLDS: RECENT PROGRESS CRAIG R. GUILBAULT Abstract. In this note we describe some recent work on ends of manifolds. In particular, we discuss progress on two different approaches to generalizing Sieben- mann’s thesis to include manifolds with non-stable fundamental groups at infinity. 1. Introduction In this note we discuss some of our recent work on ends of manifolds. For simplicity we focus our attention on one-ended open manifolds. • A manifold M n is open if it is noncompact and has no boundary. • A subset V of M n is a neighborhood of infinity if M n - V is compact. • M n is one-ended if each neighborhood of infinity contains a connected neigh- borhood of infinity. Example 1. R n is an open n-manifold for all n ≥ 1. If n ≥ 2, then R n is one-ended. Example 2. If P n is a closed connected manifold, then P n × R k is an open manifold for all n ≥ 1. P n × R k is one-ended iff k ≥ 2. Example 3. Let P n be a compact manifold with non-empty connected boundary. Then int (P n ) is a one-ended open manifold. A natural question to ask about open manifolds is the following. Question. When is an open n-manifold M n just the interior of a compact manifold with boundary? Equivalent Question. When does M n contain an “open collar” neighborhood of infinity? (V is an open collar if V ≈ ∂V × [0, 1). These questions were answered (in high dimensions) by Siebenmann in his 1965 Ph.D. thesis. Theorem 1.1 (see [Si]). A one ended open n-manifold M n (n ≥ 6) contains an open collar neighborhood of infinity if and only if each of the following is satisfied: (1) M n is inward tame at infinity, (2) π 1 is stable at infinity, and Date : December 28, 2001. 41