Information about:


TOPOLOGICAL METHODS IN GROUP THEORY


by Ross Geoghegan


 


This book was published by Springer, New York in December 2007.


It is Vol. 243 of their series Graduate Texts in Mathematics


From the Introduction:


"This is a book about the interplay between algebraic topology and the theory of infinite discrete groups. I have written it for three kinds of readers. First, it is for graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric and homological group theory. Secondly, I am writing for group theorists who would like to know more about the topological side of their subject but who have been too long away from topology. Thirdly, I hope the book will be useful to manifold topologists, both high- and low-dimensional, as a reference source for basic material on proper homotopy and homology..."


Table of Contents


PART I: ALGEBRAIC TOPOLOGY FOR GROUP THEORY


CHAPTER 1: CW COMPLEXES AND HOMOTOPY:


1.1 Review of general topology


1.2 CW complexes


1.3 Homotopy


1.4 Maps between CW complexes


1.5 Neighborhoods and complements


CHAPTER 2: CELLULAR HOMOLOGY:


2.1 Review of chain complexes


2.2 Review of singular homology


2.3 Cellular homology: the abstract theory


2.4 The degree of a map from a sphere to itself


2.5 Orientation and incidence number


2.6 The geometric cellular chain complex


2.7 Some properties of cellular homology


2.8 Further properties of cellular homology


2.9 Reduced homology




CHAPTER 3: FUNDAMENTAL GROUP AND TIETZE TRANSFORMATIONS:


3.1 Combinatorial fundamental group, Tietze transformations, Van Kampen theorem


3.2 Combinatorial description of covering spaces


3.3 Review of (topologically defined) fundamental group


3.4 Equivalence of the two definitions of the fundamental group of a CW complex


CHAPTER 4: SOME TECHNIQUES IN HOMOTOPY THEORY:


4.1 Altering a CW complex within its homotopy type


4.2 Cell trading


4.3 Domination, mapping tori and mapping telescopes


4.4 Review of homotopy groups


4.5 Geometric proof of the Hurewicz Theorem


CHAPTER 5: ELEMENTARY GEOMETRIC TOPOLOGY:


5.1 Review of topological manifolds


5.2 Simplicial complexes and combinatorial manifolds


5.3 Regular CW complexes


5.4 Incidence numbers in simplicial complexes


PART II: FINITENESS PROPERTIES OF GROUPS


CHAPTER 6: THE BOREL CONSTRUCTION AND BASS-SERRE THEORY:


6.1 The Borel construction, stacks and rebuilding


6.2 Decomposing groups which act on trees (Bass-Serre theory)


CHAPTER 7: TOPOLOGICAL FINITENESS PROPERTIES AND DIMENSION OF GROUPS:


7.1 K(G, 1)-complexes


7.2 Finiteness properties and dimension of groups


7.3 Recognizing the finiteness properties and dimension of a group


7.4 Brown's Criterion for finiteness


CHAPTER 8: HOMOLOGICAL FINITENESS PROPERTIES OF GROUPS:


8.1 Homology of groups


8.2 Homological finiteness properties


8.3 Synthetic Morse theory and the Bestvina-Brady Theorem


CHAPTER 9: FINITENESS PROPERTIES OF SOME IMPORTANT GROUPS:


9.1 Finiteness properties of Coxeter groups


9.2 Thompson's Group F and homotopy idempotents


9.3 Finiteness properties of Thompson's Group


9.4 Thompson's simple group T


9.5 The outer automorphism group of a free group


PART III: LOCALLY FINITE ALGEBRAIC TOPOLOGY FOR GROUP THEORY


CHAPTER 10: LOCALLY FINITE CW COMPLEXES AND PROPER HOMOTOPY:


10.1 Proper maps and proper homotopy theory


10.2 CW-proper maps


CHAPTER 11: LOCALLY FINITE HOMOLOGY:


11.1 Infinite cellular homology


11.2 Review of inverse and direct systems


11.3 The derived limit


11.4 Homology of ends


CHAPTER 12 COHOMOLOGY OF CW COMPLEXES


12.1 Cohomology based on infinite and finite (co)chains


12.2 Cohomology of ends


12.3 A special case: Orientation of pseudomanifolds and manifolds


12.4 Review of more homological algebra


12.5 Comparison of the various homology and cohomology theories


12.6 Homology and cohomology of products


PART IV: TOPICS IN THE COHOMOLOGY OF INFINITE GROUPS


CHAPTER 13: HOMOLOGICAL GROUP THEORY AND ENDS OF COVERING SPACES:


13.1 Cohomology of groups


13.2 Homology and cohomology of highly connected covering spaces


13.3 Topological interpretation of H*(G, RG)


13.4 Ends of spaces


13.5 Ends of groups and the structure of H^1(G, RG)


13.6 Proof of Stallings' Theorem


13.7 The structure of H^2(G, RG)


13.8 Asphericalization and an example of H^3(G,ZG)


13.9 Coxeter group examples of H^n(G,ZG)


13.10 The case H*(G, RG) = 0


13.11 An example of H*(G, RG) = 0


CHAPTER 14: FILTERED ENDS OF PAIRS OF GROUPS:


14.1 Filtered homotopy theory


14.2 Filtered chains


14.3 Filtered ends of spaces


14.4 Filtered cohomology of pairs of groups


14.5 Filtered ends of pairs of groups


CHAPTER 15: POINCARÉ DUALITY IN MANIFOLDS AND GROUPS:


15.1 CW manifolds and dual cells


15.2 Poincaré and Lefschetz duality


15.3 Poincaré duality groups and duality groups


PART V: HOMOTOPICAL GROUP THEORY


CHAPTER 16: THE FUNDAMENTAL GROUP AT INFINITY:


16.1 Connectedness at infinity


16.2 Analogs of the fundamental group


16.3 Necessary conditions for a free Z-action


16.4 Example: Whitehead's contractible 3-manifold


16.5 Group invariants: simple connectivity, stability and semistability of groups


16.6 Example: Coxeter groups and Davis manifolds


16.7 Free topological groups


16.8 Products and group extensions


16.9 Sample theorems on simple connectivity and semistability


CHAPTER 17: HIGHER HOMOTOPY THEORY OF GROUPS:


17.1 Higher proper homotopy


17.2 Higher connectivity invariants of groups


17.3 Higher invariants of group extensions


17.4 The space of proper rays


17.5 Z-set compactifications


17.6 Compactifiability at infinity as a group invariant


17.7 Strong Shape theory


PART VI: THREE ESSAYS


CHAPTER 18: ESSAYS


18.1 l_2 Poincare duality


18.2 Quasi-isometry invariants


18.3 The Bieri-Neumann-Strebel invariant


























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