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