In this section, we compute λ(M) for any oriented closed 3-manifold M with positive rank. (The rank of a closed 3-manifold M is its first Betti number β_{1}(M).) In order to do this, we frrst give M a surgery presentation as in:

*Any oriented closed 3-manifold*M*can be obtained by surgery from a rational homology sphere*R*according to the instructions of a presentation*𝕃*such that: The linking matrix of*𝕃*is null and the components of the underlying link*L*of*𝕃*are null-homologous*.

*The link*L*has then*β_{1}(M)*components, and*

PROOF: Let β =

EP - 94 PB - Princeton University Press PY - 1996 SN - 9780691021324 SP - 81 T2 - Global Surgery Formula for the Casson-Walker Invariant. (AM-140) UR - http://www.jstor.org/stable/j.ctt7zv8pj.7 Y2 - 2021/09/17/ ER -