Invariants of boundary link cobordism
 Responsibility
 Desmond Sheiham.
 Imprint
 Providence, R.I. : American Mathematical Society, 2003.
 Physical description
 ix, 110 p. ; 26 cm.
 Series
 Memoirs of the American Mathematical Society no. 784.
Online
At the library
Science Library (Li and Ma)
Serials
Call number  Note  Status 

Shelved by Series title NO.784  Unknown 
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Description
Creators/Contributors
 Author/Creator
 Sheiham, Desmond, 1974
Contents/Summary
 Bibliography
 Includes bibliographical references (p. 105108) and index.
 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.
 (source: Nielsen Book Data)
 Publisher's summary

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 $G^{(1)^q, \mu}(\Z)$ 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.
(source: Nielsen Book Data)
Subjects
 Subjects
 Ktheory.
 Knot theory.
 Cobordism theory.
Bibliographic information
 Publication date
 2003
 Series
 Memoirs of the American Mathematical Society, 00659266 ; no. 784
 Note
 "Volume 165, number 784 (first of 4 numbers)."
 ISBN
 0821833405 (alk. paper)
 9780821833407 (alk. paper)