Andrew Nicas

Varieties of group representations and Casson’s invariant for rational homology 3-spheres

Andrew Cassons Z-valued invariant for Z-homology 3-spheres is shown to extend to a Q-valued invariant for Q-homology 3-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian SL2(C) and SU(2) representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian SL2(C) or SU(2) representation is obtained. We also derive a sum theorem for Cassons invariant with respect to toroidal splittings of a Z-homology 3-sphere. Andrew Cassons lectures at MSRI in the spring of 1985 introduced an integer valued invariant of oriented integral homology 3-spheres. This invariant, constructed by means of representation spaces, yields interesting new results in low dimensional topology. In this paper we examine the extent to which Cassons procedure for defining his invariant can be used to obtain a rational valued invariant for oriented rational homology 3-spheres. Let 7r be a finitely generated group and G a Lie group. It is well known that the set R(7, G) of all homomorphisms of 7r into G can be given the structure of an analytic set in a natural manner. If G is an algebraic group, R(7r, G) becomes an algebraic set. The closed subspace of R(7r, G) consisting of representations 7r -* G with abelian image will be denoted by A(7r, G) . Let Rn(7,, G) be the union of those components of R(7r, G) which do not meet A(7r, G). When G is understood from the context, it will be dropped from the notation. If R is a commutative ring, an R-homology 3-sphere is a closed, orientable (over Z) 3-manifold with homology isomorphic to H*(S3; R). Let H(R) be the set of oriented homeomorphism types of oriented R-homology 3-spheres. For M E 11(Z) Casson defined an integer valued invariant A(M). We briefly recall his definition (see [AM] for a comprehensive exposition of Cassons MSRI lectures). Let M = WI UF W2 be a Heegard decomposition of M, where F = & W is of genus g and let F* be F punctured once. The diagram of I Received by the editors May 19, 1987 and, in revised form, December 9, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M99, 57M05, 20F99.

An intersection homology invariant for knots in a rational homology 3-sphere

THE purpose of this paper is to define a family of computable homological invariants of knots that generalize Casson’s invariant of knots. Let K be a homologically trivia1 knot in a rational homology 3-sphere N. Given a pair of integers (n, d) with n 2 1, we define a numerical invariant & and a related polynomial invariant pJt) which depend only on (N, K, n, d mod n). The invariant An,* can be thought of as an algebraic count of the number of characters of representations of the fundamental group of the complement of K into the Lie group SU(n) which take a longitude to eznid’” times the identity. The case where n and d are not relatively prime is of most interest to us here as the relatively prime case (for fibered knots) has been treated in [4]. While we define the invariants d see Theorems 5.21 and 5.22. Furthermore, an algorithm is given for determining these polynomials. We explicitly evaluate 1,. 0; see Theorem 6.4. For fibered knots, A,,, can be computed from the intersection homology Lefschetz number of the monodromy action on the moduli space of semistable holomorphic bundles of rank n and degree d and fixed determinant over a compact Riemann surface. For n and d not relatively prime, this moduli space is typically singular. Our computation relies heavily on the theory developed by Frances Kirwan ([ 14, 15, 16, 171) for desingularizing these spaces and computing their (mid-perversity) intersection homology. The polynomial invariants which we define in

On the higher Whitehead groups of a Bieberbach group

3 are best understood in the abstract framework of “cobordism functors” developed in

Higher Lefschetz Traces and Spherical Euler Characteristics

1. The axioms for such functors are somewhat reminiscent of, albeit less restrictive than, the axioms proposed for so-called “topological quantum field theories” (see Cl]). In [6] it was shown how the Alexander polynomial arises in an elementary fashion from U(1) representations in the context of cobordism functors (see [6, Theorem 4.41). Here, this is generalized to PU(n) representations from which we can obtain polynomial invariants which, at least in the case of fibered knots, are computable in terms of data derived from the Alexander polynomial. In order to put our theory into perspective, we first review the definition of Casson’s invariant. Let M be an oriented homology 3-sphere. Let H1 and Hz be two handlebodies so

Alexander invariants for virtual knots

Let T be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if p > 2 is a prime, then the p-torsion subgroup of Wh?(r) vanishes for 0 < i < 2p - 2, where Wh?(r) is the ?th higher Whitehead group of T. The proof involves Farrell and Hsiangs structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausens algebraic if-theory of spaces.

Classifying pairs of lagrangians in a hermitian vector space

Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler. The Lefschetz number of a self-map f : X → X of a space X with finitely generated homology,

Virtual knot groups and almost classical knots

Given a virtual knot

An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups

K

The Horofunction boundary of the Heisenberg Group: The Carnot-Carathéodory metric

, we construct a group

On Wh2 of a Bieberbach group

VG_K