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Title: Ends of Complexes (Cambridge Tracts in Mathematics) by B. Hughes, A. Ranicki, B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak ISBN: 0-521-57625-3 Publisher: Cambridge University Press Pub. Date: 28 August, 1996 Format: Hardcover Volumes: 1

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Summary: An introduction to topology of CW complexes at infinity.
Comment: Compact spaces have been the most desired category to work in both in topology and analysis. This is mainly due to the control that one has over the topology of the space or over the functions defined on it. Dealing with non-compact spaces requires new tools and analysis that are more difficult to use than in the compact case. This book is one of the few that specialize in the study of non-compact spaces via the 'ends' of such spaces. Loosely speaking, the 'ends' of a topological space are the directions where the space becomes non-compact: they are the complements of arbitrarily large compact subspaces of the space. The authors use tools from both algebra and homotopy theory to deal with the 'tame' ends of manifolds and CW-complexes. The treatment is well-organized and well-motivated, and anyone interested in the role that non-compact spaces play in topology, particularly in the classification of high-dimensional compact spaces, can benefit from its study.

The book is divided into three parts, the first dealing with the 'topology at infinity' using homotopy theory; the second with 'topology over the real line' for tame ends; while the third deals with the algebraic theory of ends, and the connection with algebraic K- and L-theory. The main results of the book, dealing with CW complex bands and ribbons, appear in the last three chapters of part 2, but because of space limitations, only the first part will be reviewed here.

After an introduction and chapter summary in the first sections of the book, the authors begin a rigorous study of end spaces in chapter 1. The end space e(W) of a space W is studied in terms of its homotopy type, with its path components related to the number of ends of W, and its fundamental group related to the fundamental group at infinity of W. As expected, the end space of a compact space is empty. e(W) can be simple, such as the real line, which has two ends, while the dyadic tree has uncountably many ends.

In chapter 2 the authors use direct and inverse systems of groups to study e(W), showing the connection between its weak homotopy type and an inverse system of subspaces of W, and how its homotopy groups fit into a short exact sequence involving the derived limit. Particularly interesting is their discussion on the how the homology of non-compact spaces is related to the localization and completion of rings.

The 'homology at infinity' of a compact space is studied in chapter 3, with the authors giving detailed constructions of 'locally finite' homology. The extent to which the homology at infinity is nonzero gives a measure of the non-compactness of W. The authors relate the homology groups of the end space to the homology at infinity of W, and show that it is an isomorphism if W is 'forward tame'. This is followed in chapter 4 by a discussion of cellular homology, the authors relating the singular locally finite homology groups of a certain CW complex to the cellular locally finite homology groups, generalizing the classical result.

The homology of covering spaces of W is studied in chapter 5, the authors relating locally finite homology isomorphisms of universal covers to proper homotopy equivalences, generalizing the classical Whitehead theorem. The ordinary and locally finite homology, and the homology at infinity are related to the Wall finiteness obstruction and Whitehead torsion in chapter 6. Algebraic K-theory makes its first appearance here, along with the (locally-finite) projective class, which becomes the Euler characteristic for some CW-complexes. This is followed by discussions of forward and backward tameness in chapters 7 and 8. These notions rely on the behavior of proper maps and homotopies on certain subspaces of W. In chapter 9 the authors give criteria for when a space (an absolute neighborhood retract) is forward and reverse tame, namely it must be bounded homotopy equivalent at infinity to a product with [0, infinity). Then in chapter 10, the projective class at infinity is used as an obstruction to the reverse collaring of W, and the authors prove that for the case of a manifold, the end is forward tame if and only if it is reverse tame, for certain conditions on the fundamental group. The Wall finiteness obstruction makes its appearance here, in that the projective class at infinity is its image under certain conditions on W. The case of infinite torsion in a proper homotopy equivalence of locally finite CW complexes is defined as an element of the infinite Whitehead group in chapter 11. The author uses this notion to prove an analog of the classical Siebenmann result on tame ends and collaring, namely that ends are collared if and only if they are tame and the K0 obstruction vanishes. The authors prove in partcicular that for a (strongly) locally finite CW complex W that is forward tame, the product of W with the unit circle is (infinite) simple homotopy equivalent to a forward collared CW complex (same holds true for reverse tame). That forward tameness is a 'homotopy pushout' is proven in chapter 12, at least for metric spaces that are sigma-compact. Used in the proof of this is a special notion of a mapping cylinder called a 'teardrop' mapping cylinder.