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Extra resources for Casson’s invariant for oriented homology 3-spheres : an exposition.

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R(G) is 2-connected, we have the natural isomorphism: In addition, we have the commutative diagram: For each element we have a natural evaluation map: 21 then the We have the following identity: Likewise, since Let M is 2-connected, we have the natural isomorphism: be the "dual" to the top class z of S'. ) Using the above observations, we obtain a natural homomorphism: If X is an endomorphism of then we have the following identity: The next proposition follows immediately. 1: Suppose X is an endomorphism of diagram commutes: 22 G.

We reserve the term dual for Poincare dual. 3, and these observations, we deduce the following proposition. 4: Suppose g If X is an endomorphism of G, then is a generator of G (an element of a basis of G). Let: Choosing a basis for G which contains g, we can order the basis so that: With respect to the associated identification, we obtain an homeomorphism: Hence, y(g) determines a 3n - 3 cycle in to sign. ) CHAPTER II: HEBGARD DECOMPOSITIONS AND STABLE BQUIVALBNCE 1. Heegard decompositions and models (a) The standard handlebody W Let: W = standard (model) handlebody of genus g (g » 1) F = 3W = boundary of W D = embedded 2-disk in F 0 = basepoint of 3D F on F* = F \ interior (D) , S1 = 30 .

1: Proof: (1) This is immediate from the definitions. (2) Since the conjugate of a reducible representation is clearly reducible, part (1) implies that: It remains to show that a reducible representation is conjugate to a diagonal representation. Suppose, therefore, that we have a reducible representation: reducible. 1 (2) yields an element Bquivalently, C V in of invariant. The such that: D The next corollary follows immediately from the definitions. 1 and the naturality of the action. 1(2) is also natural.

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