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**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.

\(\lambda\) (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 |