@inbook{10.2307/j.ctt7zv8pj.6, ISBN = {9780691021324}, URL = {http://www.jstor.org/stable/j.ctt7zv8pj.6}, abstract = {§4.2 to §4.5 prove the surgery formula T2 satisfied by λ, that is:Proposition T2:For any rational homology sphereM,and for any surgery presentationℍinM,the surgery formulaF(M,ℍ)is satisfied:$\text{(F(M,}\mathbb{H}\text{)) }\!\!\lambda\!\!\text{ (}{{\text{ }\!\!\chi\!\!\text{ }}_{\text{M}}}\text{(}\mathbb{H}\text{))=}\frac{\left| {{\text{H}}_{\text{1}}}\text{(}{{\text{ }\!\!\chi\!\!\text{ }}_{\text{M}}}\text{(}\mathbb{H}\text{))} \right|}{\left| {{\text{H}}_{\text{1}}}\text{(M)} \right|}\text{ }\!\!\lambda\!\!\text{ (M)+}{{\text{F}}_{\text{M}}}\text{(}\mathbb{H}\text{)}$Section 4.7 relates the one-component surgery formula to the Walker formula, recalled in §4.6. This relationship implies that λ satisfies Property T5.0:Property T5.0:IfM isa rational homology sphere,and ifλwdenotes the Walker invariant as described in[W]:$\text{ }\!\!\lambda\!\!\text{ (M)=}\frac{\left| {{\text{H}}_{\text{1}}}\text{(M)} \right|}{\text{2}}{{\text{ }\!\!\lambda\!\!\text{ }}_{\text{W}}}\text{(M)}$By definition, Proposition T2 holds for all surgery presentations in S3. Our approach to the proof of Proposition T2 will be to}, bookauthor = {Christine Lescop}, booktitle = {Global Surgery Formula for the Casson-Walker Invariant. (AM-140)}, pages = {60--80}, publisher = {Princeton University Press}, title = {The formula for surgeries starting from rational homology spheres}, year = {1996} }