Infinite Homotopy Theory: (6 K-Monographs in Mathematics)

Compactness in topology and finite generation in algebra are nice properties to start with. However, the study of compact spaces leads naturally to non-compact spaces and infinitely generated chain complexes; a classical example is the theory of covering spaces. In handling non-compact spaces we must take into account the infinity behaviour of such spaces. This necessitates modifying the usual topological and algebraic cate­ gories to obtain "proper" categories in which objects are equipped with a "topologized infinity" and in which morphisms are compatible with the topology at infinity. The origins of proper (topological) category theory go back to 1923, when Kere­ kjart6 [VT] established the classification of non-compact surfaces by adding to orien­ tability and genus a new invariant, consisting of a set of "ideal points" at infinity. Later, Freudenthal [ETR] gave a rigorous treatment of the topology of "ideal points" by introducing the space of "ends" of a non-compact space. In spite of its early ap­ pearance, proper category theory was not recognized as a distinct area of topology until the late 1960's with the work of Siebenmann [OFB], [IS], [DES] on non-compact manifolds.

Table of Contents:
I. Foundations of homotopy theory and proper homotopy theory.- § 1 Compactifications and compact maps.- § 2 Homotopy.- § 3 Categories with a cylinder functor.- § 4 Cofibration categories and homotopy theory in I-categories.- § 5 Tracks and cylindrical homotopy groups.- § 6 Homotopy groups.- § 7 Cofibres.- Appendices.- § 8 Appendix. Compact maps.- § 9 Appendix. The Freudenthal compactification.- II. Trees and spherical objects in the category Topp of compact maps.- § 1 Locally finite trees and Freudenthal ends.- Appendix. Halin’s tree lemma.- § 2 Unions in Topp.- Appendix. The proper Hilton—Milnor theorem.- § 3 Spherical objects and homotopy groups in Topp.- §4 The homotopy category of n-dimensional spherical objects in Topp.- Appendix. Classification of spherical objects under a tree.- III. Tree-like spaces and spherical objects in the category End of ended spaces.- §1 Tree-like spaces in End.- § 2 Unions in End.- § 3 Spherical objects and homotopy groups in End.- §4 The homotopy category of n-dimensional spherical objects in End.- Appendix. Classification of spherical objects under a tree-like space.- § 5 Z-sets and telescopes.- § 6 ARZ-spaces.- IV. CW-complexes.- § 1 Relative CW-complexes in Top.- § 2 Strongly locally finite CW-complexes.- § 3 Relative CW-complexes in Topp.- § 4 Relative CW-complexes in End.- § 5 Normalization of CW-complexes.- § 6 Push outs of CW-complexes.- § 7 The Blakers—Massey theorem.- § 8 The proper Whitehead theorem.- V. Theories and models of theories.- § 1 Theories of cogroups and Van Kampen theorem for proper fundamental groups.- § 2 Additive categories and additivization.- § 3 Rings associated to tree-like spaces.- § 4 Inverse limits of gr(T)-models.- § 5 Kernels in ab(T).- VI. T-controlled homology.-§ 1 R-modules and the reduced projective class group.- § 2 Chain complexes in ringoids and homology.- § 3 Cellular T-controlled homology.- § 4 Coefficients for T-controlled homology and cohomology.- § 5 The Hurewicz theorem in End.- §6 The proper homological Whitehead theorem (the 1-connected case).- § 7 Proper finiteness obstructions (the 1-connected case).- VII. Proper groupoids.- § 1 Filtered discrete objects.- § 2 The fundamental groupoid of ended spaces.- § 3 The proper homotopy category of 1-dimensional reduced relative CW-complexes.- § 4 Free D-groupoids under G.- § 5 The proper fundamental groupoid of a 1-dimensional reduced relative CW-complex.- § 6 Simplicial objects in proper homotopy theory.- VIII. The enveloping ringoid of a proper grou-poid.- § 1 The homotopy category of 1-dimensional spherical objects under T.- § 2 The ringoid S (X, T) associated to a pair (X, T) in End.- § 3 The enveloping ringoid of the proper fundamental group.- § 4 The enveloping ringoid of the proper fundamental groupoid.- IX. T-controlled homology with coefficients.- §1 The T-controlled twisted chain complex of a relative CW-complex (X, T).- § 2 The T-controlled twisted chain complex of a CW-complex X.- § 3 T-controlled cohomology and homology with local coefficients.- § 4 Proper obstruction theory.- § 5 The twisted Hurewicz homomorphism and the twisted ?-sequence in ?End.- § 6 The proper homological Whitehead theorem (the 0-connected case).- § 7 Proper finiteness obstructions (the 0-connected case).- X. Simple homotopy types with ends.- § 1 The torsion group Kl.- § 2 Simple equivalences and proper equivalences.- § 3 The topological Whitehead group.- § 4 The algebraic Whitehead group.- § 5 The proper algebraic Whitehead group.- List of symbols.

Review :
From the reviews: "In this book the authors try to deal with more general spaces in a fundamental way by setting up algebraic topology in an abstract categorical context which encompasses not only the usual category of topological spaces and continuous maps, but also several categories related to proper maps. … all concepts are carefully explained and detailed references for the proofs are given. … a good understanding of the basics of ordinary homotopy theory is all that is needed to enjoy reading this book." (F. Clauwens, Nieuw Archief voor Wiskunde, Vol. 7 (2), 2006)