Ebook: Global Surgery Formula for the Casson-Walker Invariant.

3. In Global Surgery Formula for the Casson-Walker Invariant, a function F of framed links in S

3 is described, and it is proven that F consistently defines an invariant, lamda ( l ), of closed oriented 3-manifolds. l is then expressed in terms of previously known invariants of 3-manifolds. For integral homology spheres, l is the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3-dimensional topology. l becomes simpler as the first Betti number increases.

As an explicit function of Alexander polynomials and surgery coefficients of framed links, the function F extends in a natural way to framed links in rational homology spheres. It is proven that F describes the variation of l under any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.