Search

We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere., 34 pages

THIS work forms part of our on-going effort to classify the set of concordance classes of links.RecallthatalinkL= (K,,. . . ,K,}inS “+’ is a locally flat piecewise-linear, oriented submanifold of S”+2 of which each component Ki is homeomorphic to S”. The exterior E(L) of a link L is the closure of the complement of a small regular neighborhood N(L) of L. A longitude of a component Ki is a parallel of Ki lying on the boundary of the tubular neighborhood (untwisted if n = 1). A meridian pi is a path from a basepoint to JN(L) which traverses a fiber of 8N(L) and returns. A Seifert Surface for Ki is a connected, compact, oriented, (n + 1)-manifold K E E(L) such that aK is a longitude of Ki. Links Lo, L1 are concordant (or cobordant) if there is a smooth, oriented submanifold C = {C,, . . . , C,} of S n+2 x [0, l] which meets the boundary transversely in X, is piecewise-linearly homeomorphic to L,, x [0, 11, and meets Sn+2 x {i} in Li for i = 0, 1. In the mid-603 M. Kervaire and J. Levine gave an algebraic classification of knot concordance groups (m = 1) in high dimensions (n > 1) [l]. For even n these are trivial and for odd n they are infinitely generated, being isomorphic to certain Witt groups obtained from information garnered from the Seifert surface. Extending Levine’s knot cobordism classification to links is difficult for several reasons. Firstly, if m > 1, the natural operation of connected-sum is not well-defined on concordance classes so there is no obvious group structure. Secondly, the Seifert surfaces for different components of a link may intersect, obstructing at least the naive generalization of the Seifert form information. However, the techniques do extend well to the class of boundary links. A boundary link is one which admits a collection of m disjoint Seifert surfaces, one for each component. In fact, S. Cappell and J. Shaneson classified boundary links modulo boundary link cobordism in 1980 using their homology surgery groups, followed later by Ki Ko and W. Mio who accomplished this via Seifert surfaces [2-41. A boundary link cobordism is a cobordism C between Lo and L1 for which there exist disjointly embedded 2n-manifolds IV= {IV,, . . . , ZVm) in E(C) such that IVn(S”+2 x {i}) is a system of Seifert surfaces for the boundary link Li, i = 0, 1, and such that