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

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Topology and Geometric Group Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 184))

Abstract

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.

This project was aided by a Simons Foundation Collaboration Grant.

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Notes

  1. 1.

    Despite our affinity for noncompact spaces, we are not opposed to the practice of compactification, provided it is done in a (geometrically) sensitive manner.

  2. 2.

    A proper metric space is one in which every closed metric ball is compact.

  3. 3.

    No expertise in cosmology is being claimed by the author. This description of space-time is intended only to motivate discussion.

  4. 4.

    A connected space X is aspherical if \(\pi _{k}\left( X\right) =0\) for all \(k\ge 2\).

  5. 5.

    An action by \(\Gamma \) on X is proper if, for each compact \(K\subseteq X\) at most finitely many \(\Gamma \)-translates of K intersect K. The action is cocompact if there exists a compact C such that \(\Gamma C=X\).

  6. 6.

    Sometimes closed neighborhood of infinity are preferable; then we let \(U_{i}=\overline{X-K_{i}}\). In many cases the choice is just a matter of personal preference.

  7. 7.

    Yes, this is our third distinct mathematical use of the word proper!.

  8. 8.

    The prefix “pro” is derived from “projective”. Some authors refer to inverse sequences and inverse limits as projective sequences and projective limits, respectively.

  9. 9.

    We are not being entirely forthright here. In the literature, pro-Groups usually refers to a larger category consisting of “inverse systems” of groups indexed by arbitrary partially ordered sets. We have described a subcategory, Tow-Groups, made up of those objects indexed by the natural numbers—also known as “towers”.

  10. 10.

    A complete proof would do this while keeping a base point of the loop on a base ray r.

  11. 11.

    Definitions of free product with amalgamation and HNN extension can be found in [45, 85], or any text on combinatorial group theory.

  12. 12.

    By definition, \(\underleftarrow{\lim }\left\{ K_{i} ,f_{i}\right\} \) is viewed as a subspace of the infinite product space \(\prod _{i=0}^{\infty }K_{i}\) and is topologized accordingly.

  13. 13.

    The definition of derived limit can be generalized to include nonableian groups (see [45, Sect. 11.3]), but that is not needed here.

  14. 14.

    Bestvina informally introduced the definition of weak \(\mathscr {Z}\)-structure in [9], where he also commented on his decision to omit Condition (e) from the definition of \(\mathscr {Z}\)-structure. Farrell and Lafont introduced the term \(E\mathscr {Z}\)-structure in [35].

  15. 15.

    Added in proof. An affirmative answer to this question was recently obtained by Molly Moran.

  16. 16.

    All homology here is with \(\mathbb {Z}\)-coefficients. With the same strategy and an arbitrary coefficient ring, we can also define R-homology manifold and R-homology manifold with boundary.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201, USA

    Craig R. Guilbault

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  1. Department of Mathematics, Ohio State University, Columbus, Ohio, USA

    Michael W. Davis

  2. Department of Mathematics, Ohio State University, Columbus, Ohio, USA

    James Fowler

  3. Department of Mathematics, Ohio State University Department of Mathematics, Columbus, Ohio, USA

    Jean-François Lafont

  4. Mathematical Sciences, University of Southampton, Southampton, United Kingdom

    Ian J. Leary

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Guilbault, C.R. (2016). Ends, Shapes, and Boundaries in Manifold Topology and Geometric Group Theory. In: Davis, M., Fowler, J., Lafont, JF., Leary, I. (eds) Topology and Geometric Group Theory. Springer Proceedings in Mathematics & Statistics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-43674-6_3

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