Craig R. Guilbault

Ends, Shapes, and Boundaries in Manifold Topology and Geometric Group Theory

This survey/expository article covers a variety of topics related to the “topology at infinity” of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard tools of algebraic topology. One approach uses end invariants, such as the number of ends or the fundamental group at infinity. Another approach seeks nice compactifications, then analyzes the boundaries. A thread connecting the two approaches is shape theory. In these notes we provide a careful development of several topics: homotopy and homology properties and invariants for ends of spaces, proper maps and homotopy equivalences, tameness conditions, shapes of ends, and various types of \(\mathscr {Z}\)-compactifications and \(\mathscr {Z}\)-boundaries. Classical and current research from both manifold topology and geometric group theory provide the context. Along the way, several open problems are encountered. Our primary goal is a casual but coherent introduction that is accessible to graduate students and also of interest to active mathematicians whose research might benefit from knowledge of these topics.

Manifolds with non-stable fundamental groups at infinity, II.

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 Œ0; 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 N n with compact boundary to be a homotopy collar provided @N n ,!N n 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.

Z-COMPACTIFICATIONS OF OPEN MANIFOLDS

Abstract Suppose an open n -manifold M n may be compactified to an ANR M n so that M n −M n is a Z -set in M n . It is shown that (when n ⩾5) the double of M n along its “ Z -boundary” is an n -manifold. More generally, if M n and N n each admit compactifications with homeomorphic Z -boundaries, then their union along this common boundary is an n -manifold. This result is used to show that in many cases Z -compactifiable manifolds are determined by their Z -boundaries. For example, contractible open n -manifolds with homeomorphic Z -boundaries are homeomorphic. As an application, some special cases of a weak Borel conjecture are verified. Specifically, it is shown that closed aspherical n -manifolds ( n ≠4) having isomorphic fundamental groups which are either word hyperbolic or CAT (0) have homeomorphic universal covers.

An open collar theorem for 4-manifolds

Let M 4 be an open 4-manifold with boundary. Conditions are given under which M 4 is homeomorphic to ∂M×(0,1). Applications include a 4-dimensional weak h-cobordism theorem and a classification of weakly flat embeddings of 2-spheres in S 4 . Specific examples of (n-2)-spheres embedded in S n (including n=4) are also discussed

Topological properties of spaces admitting free group actions

In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wrights method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse sequence of finitely generated free groups. We improve upon this latter result, by showing that X must have a stable finitely generated free fundamental group at infinity. Simple examples show that a free group of any finite rank is possible. We also prove that if X (as above), admits a non-cocompact action of Z+Z by covering transformations, then X is simply connected at infinity. Corollary: Every finitely presented one-ended group G which contains an element of infinite order satisfies exactly one of the following: 1) G is simply connected at infinity; 2) G is virtually a surface group; 3) The fundamental group at infinity of G is not pro-monomorphic. Our methods also provide a quick new proof of Wrights open manifold theorem.

Weak –structures for some classes of groups

Motivated by the usefulness of boundaries in the study of ‐hyperbolic and CAT(0) groups, Bestvina introduced a general axiomatic approach to group boundaries, with a goal of extending the theory and application of boundaries to larger classes of groups. The key definition is that of a “Z ‐structure” on a group G . These Z ‐ structures, along with several variations, have been studied and existence results have been obtained for a variety of new classes of groups. Still, relatively little is known about the general question of which groups admit any of the various Z ‐structures; aside from the (easy) fact that any such G must have type F, ie, G must admit a finite K.G;1/. In fact, Bestvina has asked whether every type F group admits a Z ‐structure or at least a “weak” Z ‐structure. In this paper we prove some general existence theorems for weak Z ‐structures. The main results are as follows. Theorem A If G is an extension of a nontrivial type F group by a nontrivial type F group, then G admits a weak Z ‐structure. Theorem B If G admits a finite K.G;1/ complex K such that the G ‐action on z K

An Elementary Deduction of the Topological Radon Theorem from Borsuk–Ulam

The Topological Radon Theorem states that, for every continuous function from the boundary of a (d+1)-dimensional simplex into ℝn, there exists a pair of disjoint faces in the domain whose images intersect in ℝn. The similarity between that result and the classical Borsuk–Ulam Theorem is unmistakable, but a proof that the Topological Radon Theorem follows from Borsuk–Ulam is not immediate. In this note we provide an elementary argument verifying that implication.

Mapping swirls and pseudo-spines of compact 4-manifolds

Abstract A compact subset X of the interior of a compact manifold M is a pseudo-spine of M if M − X is homeomorphic to (∂M) × [0, ∞). It is proved that a 4-manifold obtained by attaching k essential 2-handles to a B3 × S1 has a pseudo-spine which is obtained by attaching k B2s to an S1 by maps of the form z → zn. This result recovers the fact that the Mazur 4-manifold has a disk pseudo-spine (which may then be shrunk to an arc). To prove this result, the mapping swirl (a “swirled” mapping cylinder) of a map to a circle is introduced, and a fundamental property of mapping swirls is established: homotopic maps to a circle have homeomorphic mapping swirls. Several conjectures concerning the existence of pseudo-spines in compact 4-manifolds are stated and discussed, including the following two related conjectures: every compact contractible 4-manifold has an arc pseudo-spine, and every compact contractible 4-manifold has a handlebody decomposition with no 3- or 4-handles. It is proved that an important class of compact contractible 4-manifolds described by Poenaru satisfies the latter conjecture.