An Extension of Casson's Invariant. (AM-126), Volume 126 (Annals of Mathematics Studies)

This book describes an invariant, l, of oriented rational homology 3-spheres which is a generalization of work of Andrew Casson in the integer homology sphere case. Let R(X) denote the space of conjugacy classes of representations of p(X) into SU(2). Let (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is declared to be an appropriately defined intersection number of R(W) and R(W) inside R(F). The definition of this intersection number is a delicate task, as the spaces involved have singularities. A formula describing how l transforms under Dehn surgery is proved. The formula involves Alexander polynomials and Dedekind sums, and can be used to give a rather elementary proof of the existence of l. It is also shown that when M is a Z-homology sphere, l(M) determines the Rochlin invariant of M Contents 0 Introduction 1 Topology of Representation Spaces 2 Definition of λ 3 Various Properties of λ 4 The Dehn Surgery Formula 5 Combinatorial Definition of λ 6 Consequences of the Dehn Surgery Formula A Dedekind Sums B Alexander Polynomials Bibliography