# Invariants of Boundary Link Cobordism

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*Desmond Sheiham*

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
odd-dimensional \(F_\mu\)-links represent the same cobordism class
in \(C_{2q-1}(F_\mu)\) assuming \(q>1\). We proceed to compute
the isomorphism class of \(C_{2q-1}(F_\mu)\), generalizing Levine's
computation of the knot cobordism group \(C_{2q-1}(F_1)\).

Our starting point is the algebraic formulation of Levine, Ko
and Mio who identify \(C_{2q-1}(F_\mu)\) 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
torsion-free 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 self-dual
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 finite-dimensional 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 well-developed. There are five classes
of algebras to be considered; complete Witt invariants are available
for four classes, those for which the local-global principle applies.
An algebra in the fifth class, namely a quaternion algebra with non-standard
involution, requires an additional Witt invariant which is defined if all
the local invariants vanish.

#### Table of Contents

# Table of Contents

## Invariants of Boundary Link Cobordism

- Contents v6 free
- Acknowledgements ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Main Results 1728
- Chapter 3. Preliminaries 2940
- Chapter 4. Morita Equivalence 4152
- Chapter 5. Devissage 4758
- Chapter 6. Varieties of Representations 5162
- Chapter 7. Generalizing Pfister's Theorem 5970
- Chapter 8. Characters 6374
- Chapter 9. Detecting Rationality and Integrality 6778
- Chapter 10. Representation Varieties: Two Examples 7586
- Chapter 11. Number Theory Invariants 8192
- Chapter 12. All Division Algebras Occur 91102
- Appendix I. Primitive Element Theorems 99110
- Appendix II. Hermitian Categories 101112
- Bibliography 105116
- Index 109120