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Combinatorial Methods in Topology and Algebraic Geometry
Edited by: John R. Harper and Richard Mandelbaum

Contemporary Mathematics
1985; 349 pp; softcover
Volume: 44
Reprint/Revision History:
reprinted 1988
ISBN-10: 0-8218-5039-3
ISBN-13: 978-0-8218-5039-8
List Price: US$56
Member Price: US$44.80
Order Code: CONM/44
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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, 3-manifolds, 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 Andrews-Curtis-Conjecture and related problems." In addition, J. M. Montesino has provided summary articles on both 3- and 4-manifolds.

Table of Contents

Topology and Combinatorial Group Theory
  • M. Cohen, W. Metzler, and K. Sauermann -- Collapses of \(K\times I\) and group presentations
  • W. Metzler -- On the Andrews-Curtis-Conjecture and related problems
  • M. Lustig and W. Metzler -- Integral representations of \(AutF^n\) and presentation classes of groups
  • R. Goldstein and E. C. Turner -- A note on commutators and squares in free products
  • K. B. Lee and F. Raymond -- Rigidity of almost crystallographic groups
  • J. R. Stallings -- Finite graphs and free groups
  • C. Tretkoff and M. Tretkoff -- A topological proof of a theorem of Brunner and Burns about M. Hall groups
Knot Theory
  • D. Gabai -- The Murasugi sum is a natural geometric operation II
  • L. H. Kauffman -- The Arf Invariant of classical knots
  • W. B. R. Lickorish -- The unknotting number of a classical knot
  • B. Trace -- A general position theorem for surfaces in Euclidean \(4\)-space
  • A. L. Edmonds -- n the equivariant Dehn lemma
  • J. Hempel -- Virtually Haken manifolds
  • J. M. Montesinos -- Lectures on \(3\)-fold simple coverings and \(3\)-manifolds
  • H. S. Oh -- The Witt classes of Seifert manifolds
  • M. Scharlemann -- Outermost forks and a theorem of Jaco
  • J. R. Stallings -- Surfaces in \(3\)-manifolds
Homotopy Theory and Infinite Dimensional Topology
  • M. G. Barratt -- Taming Hopf invariants
  • F. R. Cohen -- Artin's braid groups and classical homotopy theory
  • L. R. Rubin -- More compacta of infinite cohomological dimension
  • A. Zabrodsky -- Endomorphisms in the homotopy category
Four Manifolds and Algebraic Surfaces
  • S. Akbulut -- On fake \(S^3\sim \times S^1 \# S^2\times S^2\)
  • N. Goldstein -- Manifolds having non-ample Norman bundles in quadrices
  • R. Mandelbaum -- Lefschetz fibrations of Riemann surfaces and decompositions of complex elliptic surfaces
  • B. Moishezon -- Algebraic surfaces and the arithmetic of braids, II
  • J. Montesinos -- A note on moves and on irregular coverings of \(S^4\)
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