Casson's invariant and surgery on knots

Abstract

We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).

Fixed points of the mapping class group in the $SU(n)$ moduli spaces

“…We will also see that our technique can show that for all n ≥ 2 there is some integer m (which may depend on n) and a point in M(n) which is preserved by the m'th power of any f in Γ. In some particular cases, a positive answer to question 1 was obtained by Frohman in [2] (see also [3]). By using some involved cohomology calculations on the classifying space of certain gauge groups, Frohman gives an existence proof for fixed points for certain diffeomorphism of surfaces with 1 boundary component.…”

Invariants for Homology 3-Spheres

“…An 5U(n) Casson invariant of knots was defined in [102] and [103] as an extension of the Casson invariant A' (k c 17), see Section 3.1. It can be viewed as an algebraic-topological count of the number of characters of 5U(n) representations of the knot group which take a longitude into a given conjugacy class.…”

An orientation for the SU(2)-representation space of knot groups

“…Even if ∆ k (1) = 0 the knot k ⊂ S 3 might still have Property P and in this case the SU(2)-representations might still be useful for proving Property P (see [Bur90,FL92]). A first step in the program of generalizing Burde's proof of Property P for 2-bridge knots (see [Bur90]) is to find knots with a non-trivial SU(2)-representation space.…”