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: Collapses of $K\times I$ and group presentations by M. Cohen, W. Metzler, and K. Sauermann On the Andrews-Curtis-Conjecture and related problems by W. Metzler Integral representations of $AutF^n$ and presentation classes of groups by M. Lustig and W. Metzler A note on commutators and squares in free products by R. Goldstein and E. C. Turner Rigidity of almost crystallographic groups by K. B. Lee and F. Raymond Finite graphs and free groups by J. R. Stallings A topological proof of a theorem of Brunner and Burns about M. Hall groups by C. Tretkoff and M. Tretkoff Knot Theory: The Murasugi sum is a natural geometric operation II by D. Gabai The Arf Invariant of classical knots by L. H. Kauffman The unknotting number of a classical knot by W. B. R. Lickorish A general position theorem for surfaces in Euclidean $4$-space by B. Trace 3-Manifolds: n the equivariant Dehn lemma by A. L. Edmonds Virtually Haken manifolds by J. Hempel Lectures on $3$-fold simple coverings and $3$-manifolds by J. M. Montesinos The Witt classes of Seifert manifolds by H. S. Oh Outermost forks and a theorem of Jaco by M. Scharlemann Surfaces in $3$-manifolds by J. R. Stallings Homotopy Theory and Infinite Dimensional Topology: Taming Hopf invariants by M. G. Barratt Artin's braid groups and classical homotopy theory by F. R. Cohen More compacta of infinite cohomological dimension by L. R. Rubin Endomorphisms in the homotopy category by A. Zabrodsky Four Manifolds and Algebraic Surfaces: On fake $S^3\sim \times S^1 \ No. S^2\times S^2$ by S. Akbulut Manifolds having non-ample Norman bundles in quadrices by N. Goldstein Lefschetz fibrations of Riemann surfaces and decompositions of complex elliptic surfaces by R. Mandelbaum Algebraic surfaces and the arithmetic of braids, II by B. Moishezon A note on moves and on irregular coverings of $S^4$ by J. Montesinos.