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This book describes an invariant, l, of oriented rational homology 3-spheres which is a generalization of work of Andrew Casson in the integer homology sphere case. Let R(X) denote the space of conjugacy classes of representations of p(X) into SU(2). Let (W,W,F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is declared to be an appropriately defined intersection number of R(W) and R(W) inside R(F). The definition of this intersection number is a delicate task, as the spaces involved have singularities. A formula describing how l transforms under Dehn surgery is proved. The formula involves Alexander polynomials and Dedekind sums, and can be used to give a rather elementary proof of the existence of l. It is also shown that when M is a Z-homology sphere, l(M) determines the Rochlin invariant of M.

Three-manifolds (Topology) --- Invariants. --- Absolute value. --- Andrew Casson. --- Basis (linear algebra). --- Cohomology. --- Dan Freed. --- Dehn surgery. --- Dehn twist. --- Determinant. --- Diagram (category theory). --- Disk (mathematics). --- Elementary proof. --- Fundamental group. --- General position. --- Heegaard splitting. --- Homology sphere. --- Identity matrix. --- Inner product space. --- Lie group. --- Mathematical sciences. --- Morris Hirsch. --- Normal bundle. --- Scientific notation. --- Sequence. --- Surjective function. --- Symplectic geometry. --- Theorem. --- Topology.

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There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields. These papers are dedicated to the memory of Ralph H. Fox, one of the world's leading topologists, by colleagues, former students, and friends.In knot theory, papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten. Of these several are devoted to the study of branched covering spaces over knots and links, while others utilize the braid groups of Artin.Cossey and Smythe, Stallings, and Strasser address themselves to group theory. In his contribution Stallings describes the calculation of the groups In/In+1 where I is the augmentation ideal in a group ring RG. As a consequence, one has for each k an example of a k-generator l-relator group with no free homomorphs. In the third part, papers by Birman, Cappell, Milnor, Montesinos, Papakyriakopoulos, and Shalen comprise the treatment of 3-manifolds. Milnor gives, besides important new results, an exposition of certain aspects of our current knowledge regarding the 3- dimensional Brieskorn manifolds.

Knot theory. --- Group theory. --- Three-manifolds (Topology) --- 3-manifold. --- 3-sphere. --- Additive group. --- Alexander duality. --- Algebraic equation. --- Algebraic surface. --- Algebraic variety. --- Automorphic form. --- Automorphism. --- Big O notation. --- Bilinear form. --- Borromean rings. --- Boundary (topology). --- Braid group. --- Cartesian product. --- Central series. --- Chain rule. --- Characteristic polynomial. --- Coefficient. --- Cohomological dimension. --- Commutative ring. --- Commutator subgroup. --- Complex Lie group. --- Complex coordinate space. --- Complex manifold. --- Complex number. --- Conjugacy class. --- Connected sum. --- Coprime integers. --- Coset. --- Counterexample. --- Cyclic group. --- Dedekind domain. --- Diagram (category theory). --- Diffeomorphism. --- Disjoint union. --- Divisibility rule. --- Double coset. --- Equation. --- Equivalence class. --- Euler characteristic. --- Fiber bundle. --- Finite group. --- Fundamental group. --- Generating set of a group. --- Graded ring. --- Graph product. --- Group ring. --- Groupoid. --- Heegaard splitting. --- Holomorphic function. --- Homeomorphism. --- Homological algebra. --- Homology (mathematics). --- Homology sphere. --- Homomorphism. --- Homotopy group. --- Homotopy sphere. --- Homotopy. --- Hurewicz theorem. --- Infimum and supremum. --- Integer matrix. --- Integer. --- Intersection number (graph theory). --- Intersection theory. --- Knot group. --- Knot polynomial. --- Loop space. --- Main diagonal. --- Manifold. --- Mapping cylinder. --- Mathematical induction. --- Meromorphic function. --- Monodromy. --- Monomorphism. --- Multiplicative group. --- Permutation. --- Poincaré conjecture. --- Principal ideal domain. --- Proportionality (mathematics). --- Quotient space (topology). --- Riemann sphere. --- Riemann surface. --- Seifert fiber space. --- Simplicial category. --- Special case. --- Spectral sequence. --- Subgroup. --- Submanifold. --- Surjective function. --- Symmetric group. --- Symplectic matrix. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Torus knot. --- Triangle group. --- Variable (mathematics). --- Weak equivalence (homotopy theory).