Infinite Homotopy Theory

2001
ISBN: 978-0-7923-6982-0
Verlag: Springer Netherlands

Buch, Englisch, Band 6, 296 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1350 g

Reihe: 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.

Zielgruppe

Research

Autoren/Hrsg.

Quintero, A.

Baues, H-J.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Topologie Analytische Topologie
• Mathematik | Informatik Mathematik Topologie Algebraische Topologie
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie

Inhaltsverzeichnis

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.