UID:
almafu_9959239102702883
Format:
1 online resource (xi, 412 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88487-5
,
1-107-36684-4
,
1-107-37151-1
,
1-107-36193-1
,
1-107-36902-9
,
1-299-40458-8
,
1-107-36438-8
,
0-511-62935-4
Series Statement:
London Mathematical Society lecture note series ; 197
Content:
Basic work on two-dimensional homotopy theory dates back to K. Reidemeister and J. H. C. Whitehead. Much work in this area has been done since then, and this book considers the current state of knowledge in all the aspects of the subject. The editors start with introductory chapters on low-dimensional topology, covering both the geometric and algebraic sides of the subject, the latter including crossed modules, Reidemeister-Peiffer identities, and a concrete and modern discussion of Whitehead's algebraic classification of 2-dimensional homotopy types. Further chapters have been skilfully selected and woven together to form a coherent picture. The latest algebraic results and their applications to 3- and 4-dimensional manifolds are dealt with. The geometric nature of the subject is illustrated to the full by over 100 diagrams. Final chapters summarize and contribute to the present status of the conjectures of Zeeman, Whitehead, and Andrews-Curtis. No other book covers all these topics. Some of the material here has been used in courses, making this book valuable for anyone with an interest in two-dimensional homotopy theory, from graduate students to research workers.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; Contents; Editors' Preface; Addresses of Authors; I Geometric Aspects of Two-Dimensional Complexes; 1 Complexes of Low Dimensions and Group Presentations . . .; 1.1 Inductive construction of CW-complexes; 1.2 Questions of subdivision and triangulation; 1.3 Reading off presentations for TTI of a CW-complex; 1.4 PLCW-complexes; 2 Simple-Homotopy and Low Dimensions; 2.1 A survey on geometric simple-homotopy; 2.2 Some examples; 2.3 3-deformation types and (Q**-transformations; 3 P.L. Embeddings of 2-Complexes into Manifolds; 3.1 3-dimensional thickenings
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3.2 4- and 5-dimensional thickenings4 Three Conjectures and Further Problems; 4.1 (Generalized) Andrews-Curtis conjecture; 4.2 Zeeman collapsing conjecture; 4.3 Whitehead asphericity conjecture as a special problem of dimension 2; 4.4 Further open questions; II Algebraic Topology for Two Dimensional Complexes; 1 Techniques in Homotopy; 1.1 Simplicial Techniques; 1.2 Combinatorial Maps; 2 Homotopy Groups for 2-Complexes 62; 2.1 Fundamental sequence for a 2-complex K; 2.2 II(K) and the homotopy type of a 2-complex K; 3 Equivariant World for 2-Complexes; 3.1 Hurewicz Isomorphism Theorems
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3.2 Two Dimensional Equivariant World4 Mac Lane-Whitehead Algebraic Types; 4.1 Homology and Cohomology of Groups; 4.2 Maps between 2-complexes; III Homotopy and Homology Classification of 2-Complexes; 1 Bias Invariant & Homology Classification; 1.1 Bias as a homotopy obstruction; 1.2 Bias as the complete homology obstruction; 1.3 Homotopy distinction of twisted presentations; 2 Classifications for Finite Abelian TTI Ill; 2.1 The Browning obstruction group; 2.2 Homotopy classification for finite abelian TTI; 3 Classifications for Non-Finite TTI (with Cynthia Hog-Angeloni); 3.1 Infinite groups
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generalized Browning invariant3.2 Results when TTI is a free product of cyclic groups; 3.3 Trees of homotopy types, simple-homotopy types, and 3_deformation types; 3.4 Problems for Chapter III; IV Crossed Modules and n2 Homotopy Modules; 1 Introduction; 2 Crossed and Precrossed Modules; 2.1 Free crossed modules; 2.2 A characterization of free crossed modules; 2.3 Projective crossed modules; 2.4 Two-complexes and projective crossed modules; 2.5 The kernel of a projective crossed module; 3 On the Second Homotopy Module of a 2-Complex; 3.1 Coproducts of crossed modules; 3.2 A special case
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3.3 On the kernel of a coproduct of crossed modules3.4 Computing π2 from subcomplexes; 4 Identity Properties; 4.1 H-Cockcroft complexes; 4.2 Characterizing Cockcroft complexes; 4.3 Minimal subgroups; V Calculating Generators of n2; 1 The Theory of Pictures; 1.1 Pictures; 1.2 Homotopy theory of pictures; 2 Generation of Π2; 2.1 Asphericity; 2.2 A Dehn algorithm for π2; 2.3 Subpresentations; 3 Applications and Results; 3.1 Universal group of a family of subgroups; 3.2 Generalized graphs of groups; 3.3 Split extensions; 3.4 Iterative constructions; 3.5 The Steinberg group and K3 of a ring
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3.6 Presentations defined using simplicial complexes
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English
Additional Edition:
ISBN 0-521-44700-3
Language:
English
Subjects:
Mathematics
URL:
https://doi.org/10.1017/CBO9780511629358
