Manifolds with non-stable fundamental groups at infinity

Guilbault, Craig R.; Tinsley, Frederick C.

doi:10.2140/gt.2000.4.537

Cited by 20 publications

(83 citation statements)

References 23 publications

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“…(In [7] we took the traditional route and assumed M n was an open manifold; but this is unnecessary as long as @M n is compact.) Recall that a 0-neighborhood of infinity is a generalized 1-neighborhood of infinity provided By the Generalized .n 3/-neighborhoods Theorem ([7, Theorem 5]), inward tameness alone allows us to obtain a cofinal sequence fU i g of generalized .n 3/-neighborhoods of infinity in M n .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…This can be done so that each remains a generalized .n 3/-neighborhood of infinity, and so that none of the fundamental groups of the neighborhoods of infinity or their boundaries are changed. To save on notation, we continue to denote this improved collection by fU i g. See [7,Lemma 14] for details. By the finite generation of H n 2 .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Call this kernel K iC1 . By a basic theorem from combinatorial group theory (see [13] or [7,Lemma 3]) K iC1 is the normal closure of a finite collection of elements of 1 . @U iC1 /.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Manifolds with non-stable fundamental groups at infinity, III

Guilbault

Tinsley

2006

Geom. Topol.

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We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6. 57N15, 57Q12; 57R65, 57Q10

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Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Call this kernel K iC1 . By a basic theorem from combinatorial group theory (see [13] or [7,Lemma 3]) K iC1 is the normal closure of a finite collection of elements of 1 . @U iC1 /.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Manifolds with non-stable fundamental groups at infinity, III

Guilbault

Tinsley

2006

Geom. Topol.

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

Guilbault

2016

Springer Proceedings in Mathematics &Amp; Statistics

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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 Z-compactifications and 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

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More examples of pseudo-collars on high-dimensional manifolds

Rolland

2020

Topology and its Applications

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In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and similar fundamental group systems at infinity. Construction of that family was very specific; it relied on properties of Thompson's group V . In this paper, we provide a more general approach to constructing similar collections of examples. Instead of using Thompson's group V , we base our new examples on a broader and more common collection of groups, in particular, fundamental groups of certain hyperbolic 3-manifolds.

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Manifolds with non-stable fundamental groups at infinity

Cited by 20 publications

References 23 publications

Manifolds with non-stable fundamental groups at infinity, III

Manifolds with non-stable fundamental groups at infinity, III

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

More examples of pseudo-collars on high-dimensional manifolds

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