## Casson’s invariant for oriented homology 3-spheres. An exposition.(English)Zbl 0695.57011

In 1985 Casson introduced an invariant for homology 3-spheres that has interesting applications to low-dimensional topology. Let K be a knot in an oriented homology 3-sphere M and let $$K_ n$$ be the homology sphere obtained from M by 1/n-Dehn surgery on K (where $$K_ 0=M)$$. Casson’s invariant $$\lambda$$ (M) is the unique invariant of oriented homology 3- spheres satisfying the following two properties: (1) $$\lambda (K_{n+1)}-\lambda (K_ n)=1/2$$ (second derivative of the symmetrized Alexander polynomial of K evaluated at 1), (2) $$\lambda (S^ 3)=0$$. Since every homology sphere M can be obtaind from a sequence of $$\pm 1$$ Dehn surgeries on knots in homology spheres, starting with a knot in $$S^ 3$$, properties (1) and (2) show uniqueness of the invariant $$\lambda$$ (M). The proof of the existence occupies the six chapters of the exposition under review.
$$\lambda$$ (M) is defined in terms of the intersection product of representation spaces of the fundamental groups of the two handlebodies arising from a Heegaard splitting of M. So in Chapter 1 representation spaces (into $$S^ 3)$$ are reviewed and in Chapter 2 and 3 Heegaard splittings with their associated representation spaces are discussed. In Chapter 4 Casson’s invariant is defined and shown to be independent of the Heegaard decomposition. It follows from this definition that $$\lambda (M)=0$$ for homotopy spheres M. Also it follows that $$\lambda (-M)=- \lambda (M).$$ Chapter 5 is a study of the difference cycle $$\delta$$ associated to $$\lambda '(K)=\lambda (K_{n+1})-\lambda (K_ n)$$ and proves (modulo a technical fact exhibited in Chapter 6) property (1). In this chapter it is also shown that $$\lambda$$ (M) and the Rochlin invariant agree mod 2. In particular this implies that any homotopy 3- sphere and any amphicheiral homology 3-sphere has zero Rochlin invariant.
Reviewer: W.Heil

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes