eBook ISBN:  9781470403829 
Product Code:  MEMO/165/784.E 
List Price:  $61.00 
MAA Member Price:  $54.90 
AMS Member Price:  $36.60 
eBook ISBN:  9781470403829 
Product Code:  MEMO/165/784.E 
List Price:  $61.00 
MAA Member Price:  $54.90 
AMS Member Price:  $36.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 165; 2003; 110 ppMSC: Primary 18; 57; 16
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.
ReadershipGraduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology.

Table of Contents

Chapters

1. Introduction

2. Main results

3. Preliminaries

4. Morita equivalence

5. Devissage

6. Varieties of representations

7. Generalizing Pfister’s theorem

8. Characters

9. Detecting rationality and integrality

10. Representation varieties: Two examples

11. Number theory invariants

12. All division algebras occur


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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.
Graduate students and research mathematicians interested in algebra, algebraic geometry, geometry, and topology.

Chapters

1. Introduction

2. Main results

3. Preliminaries

4. Morita equivalence

5. Devissage

6. Varieties of representations

7. Generalizing Pfister’s theorem

8. Characters

9. Detecting rationality and integrality

10. Representation varieties: Two examples

11. Number theory invariants

12. All division algebras occur