eBook ISBN:  9780821876299 
Product Code:  CONM/44.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821876299 
Product Code:  CONM/44.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 44; 1985; 349 ppMSC: Primary 57;
This collection marks the recent resurgence of interest in combinatorial methods, resulting from their deep and diverse applications both in topology and algebraic geometry. Nearly thirty mathematicians met at the University of Rochester in 1982 to survey several of the areas where combinatorial methods are proving especially fruitful: topology and combinatorial group theory, knot theory, 3manifolds, homotopy theory and infinite dimensional topology, and four manifolds and algebraic surfaces. This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory and geometric topology, as well as to established mathematicians with interests in these areas. For both student and professional mathematicians, the book provides practical suggestions for research directions still to be explored, as well as the aesthetic pleasures of seeing the interplay between algebra and topology which is characteristic of this field.
In several areas the book contains the first general exposition published on the subject. In topology, for example, the editors have included M. Cohen, W. Metzler and K. Sauerman's article on “Collapses of \(K\times I\) and group presentations” and Metzler's “On the AndrewsCurtisConjecture and related problems.” In addition, J. M. Montesino has provided summary articles on both 3 and 4manifolds.

Table of Contents

Topology and Combinatorial Group Theory [ MR 813097 ]

M. Cohen, W. Metzler and K. Sauermann — Collapses of $K\times I$ and group presentations [ MR 813098 ]

Wolfgang Metzler — On the AndrewsCurtis conjecture and related problems [ MR 813099 ]

Martin Lustig and Wolfgang Metzler — Integral representations of ${\rm Aut}\,F^n$ and presentation classes of groups [ MR 813100 ]

Richard Z. Goldstein and Edward C. Turner — A note on commutators and squares in free products [ MR 813101 ]

Kyung Bai Lee and Frank Raymond — Rigidity of almost crystallographic groups [ MR 813102 ]

John R. Stallings — Finite graphs and free groups [ MR 813103 ]

C. L. Tretkoff and M. D. Tretkoff — A topological proof of a theorem of Brunner and Burns about M. Hall groups [ MR 813104 ]

Knot Theory [ MR 813097 ]

David Gabai — The Murasugi sum is a natural geometric operation. II [ MR 813105 ]

Louis H. Kauffman — The Arf invariant of classical knots [ MR 813106 ]

W. B. Raymond Lickorish — The unknotting number of a classical knot [ MR 813107 ]

Bruce Trace — A general position theorem for surfaces in Euclidean $4$space [ MR 813108 ]

3Manifolds [ MR 813097 ]

Allan L. Edmonds — On the equivariant Dehn lemma [ MR 813109 ]

John Hempel — Virtually Haken manifolds [ MR 813110 ]

José María Montesinos — Lectures on $3$fold simple coverings and $3$manifolds [ MR 813111 ]

Hae Soo Oh — Witt classes of torsion linking forms [ MR 813112 ]

Martin Scharlemann — Outermost forks and a theorem of Jaco [ MR 813113 ]

John R. Stallings — Surfaces in $3$manifolds [ MR 813114 ]

Homotopy Theory and Infinite Dimentional Topology [ MR 813097 ]

M. G. Barratt — Taming Hopf invariants [ MR 813115 ]

F. R. Cohen — Artin’s braid groups and classical homotopy theory [ MR 813116 ]

Leonard R. Rubin — More compacta of infinite cohomological dimension [ MR 813117 ]

A. Zabrodsky — Endomorphisms in the homotopy category [ MR 813118 ]

Four Manifolds and Algebraic Surfaces [ MR 813097 ]

Selman Akbulut — On fake $S^3\tilde \times S^1$ #$S^2\times S^2$ [ MR 813119 ]

Norman Goldstein — Manifold having nonample normal bundles in quadrics [ MR 813120 ]

Richard Mandelbaum — Lefschetz fibrations of Riemann surfaces and decompositions of complex elliptic surfaces [ MR 813121 ]

B. Moishezon — Algebraic surfaces and the arithmetic of braids. II [ MR 813122 ]

José María Montesinos — A note on moves and on irregular coverings of $S^4$ [ MR 813123 ]


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This collection marks the recent resurgence of interest in combinatorial methods, resulting from their deep and diverse applications both in topology and algebraic geometry. Nearly thirty mathematicians met at the University of Rochester in 1982 to survey several of the areas where combinatorial methods are proving especially fruitful: topology and combinatorial group theory, knot theory, 3manifolds, homotopy theory and infinite dimensional topology, and four manifolds and algebraic surfaces. This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory and geometric topology, as well as to established mathematicians with interests in these areas. For both student and professional mathematicians, the book provides practical suggestions for research directions still to be explored, as well as the aesthetic pleasures of seeing the interplay between algebra and topology which is characteristic of this field.
In several areas the book contains the first general exposition published on the subject. In topology, for example, the editors have included M. Cohen, W. Metzler and K. Sauerman's article on “Collapses of \(K\times I\) and group presentations” and Metzler's “On the AndrewsCurtisConjecture and related problems.” In addition, J. M. Montesino has provided summary articles on both 3 and 4manifolds.

Topology and Combinatorial Group Theory [ MR 813097 ]

M. Cohen, W. Metzler and K. Sauermann — Collapses of $K\times I$ and group presentations [ MR 813098 ]

Wolfgang Metzler — On the AndrewsCurtis conjecture and related problems [ MR 813099 ]

Martin Lustig and Wolfgang Metzler — Integral representations of ${\rm Aut}\,F^n$ and presentation classes of groups [ MR 813100 ]

Richard Z. Goldstein and Edward C. Turner — A note on commutators and squares in free products [ MR 813101 ]

Kyung Bai Lee and Frank Raymond — Rigidity of almost crystallographic groups [ MR 813102 ]

John R. Stallings — Finite graphs and free groups [ MR 813103 ]

C. L. Tretkoff and M. D. Tretkoff — A topological proof of a theorem of Brunner and Burns about M. Hall groups [ MR 813104 ]

Knot Theory [ MR 813097 ]

David Gabai — The Murasugi sum is a natural geometric operation. II [ MR 813105 ]

Louis H. Kauffman — The Arf invariant of classical knots [ MR 813106 ]

W. B. Raymond Lickorish — The unknotting number of a classical knot [ MR 813107 ]

Bruce Trace — A general position theorem for surfaces in Euclidean $4$space [ MR 813108 ]

3Manifolds [ MR 813097 ]

Allan L. Edmonds — On the equivariant Dehn lemma [ MR 813109 ]

John Hempel — Virtually Haken manifolds [ MR 813110 ]

José María Montesinos — Lectures on $3$fold simple coverings and $3$manifolds [ MR 813111 ]

Hae Soo Oh — Witt classes of torsion linking forms [ MR 813112 ]

Martin Scharlemann — Outermost forks and a theorem of Jaco [ MR 813113 ]

John R. Stallings — Surfaces in $3$manifolds [ MR 813114 ]

Homotopy Theory and Infinite Dimentional Topology [ MR 813097 ]

M. G. Barratt — Taming Hopf invariants [ MR 813115 ]

F. R. Cohen — Artin’s braid groups and classical homotopy theory [ MR 813116 ]

Leonard R. Rubin — More compacta of infinite cohomological dimension [ MR 813117 ]

A. Zabrodsky — Endomorphisms in the homotopy category [ MR 813118 ]

Four Manifolds and Algebraic Surfaces [ MR 813097 ]

Selman Akbulut — On fake $S^3\tilde \times S^1$ #$S^2\times S^2$ [ MR 813119 ]

Norman Goldstein — Manifold having nonample normal bundles in quadrics [ MR 813120 ]

Richard Mandelbaum — Lefschetz fibrations of Riemann surfaces and decompositions of complex elliptic surfaces [ MR 813121 ]

B. Moishezon — Algebraic surfaces and the arithmetic of braids. II [ MR 813122 ]

José María Montesinos — A note on moves and on irregular coverings of $S^4$ [ MR 813123 ]