Memoirs of the American Mathematical Society 2003; 110 pp; softcover Volume: 165 ISBN10: 0821833405 ISBN13: 9780821833407 List Price: US$58 Individual Members: US$34.80 Institutional Members: US$46.40 Order Code: MEMO/165/784
 An \(n\)dimensional \(\mu\)component boundary link is a codimension \(2\) embedding of spheres \(L=\sqcup_{\mu}S^n \subset S^{n+2}\) such that there exist \(\mu\) disjoint oriented embedded \((n+1)\)manifolds which span the components of \(L\). An \(F_\mu\)link is a boundary link together with a cobordism class of such spanning manifolds. The \(F_\mu\)link cobordism group \(C_n(F_\mu)\) is known to be trivial when \(n\) is even but not finitely generated when \(n\) is odd. Our main result is an algorithm to decide whether two odddimensional \(F_\mu\)links represent the same cobordism class in \(C_{2q1}(F_\mu)\) assuming \(q>1\). We proceed to compute the isomorphism class of \(C_{2q1}(F_\mu)\), generalizing Levine's computation of the knot cobordism group \(C_{2q1}(F_1)\). Our starting point is the algebraic formulation of Levine, Ko and Mio who identify \(C_{2q1}(F_\mu)\) with a surgery obstruction group, the Witt group \(G^{(1)^q,\mu}(\mathbb{Z})\) of \(\mu\)component Seifert matrices. We obtain a complete set of torsionfree invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to `algebraically integral' simple selfdual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties. To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing \(G^{(1)^q,\mu}(\mathbb{Q})\) as an infinite direct sum of Witt groups of finitedimensional division \(\mathbb{Q}\)algebras with involution. The Witt group of every such algebra appears as a summand infinitely often. The theory of symmetric and hermitian forms over these division algebras is welldeveloped. There are five classes of algebras to be considered; complete Witt invariants are available for four classes, those for which the localglobal principle applies. An algebra in the fifth class, namely a quaternion algebra with nonstandard involution, requires an additional Witt invariant which is defined if all the local invariants vanish. Readership Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology. Table of Contents  Introduction
 Main results
 Preliminaries
 Morita Equivalence
 Devissage
 Varieties of representations
 Generalizing Pfister's theorem
 Characters
 Detecting rationality and integrality
 Representation varieties: Two examples
 Number theory invariants
 All division algebras occur
 Appendix I. Primitive element theorems
 Appendix II. Hermitian categories
 Bibliography
 Index
