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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 . Figure 4 We may choose a family of loops on (F*,0), as in Figure 4, such that: Let and denote the homology classes of and respectively.

R be a group. The space of representations of G in S3 is defined as follows: R(G) = Hom(G,S 3 ) = {p : G —> S 3 I ρ We give R(G) topology and S3 is a representation} . the compact open topology, where G has the discrete the usual topology. To each homomorphism X : G 1 -» G 2 , there is an associated continuous map: R(X) : R(G a ) —> R(G 1 ) ρ I—> ρ·\ . Clearly, we have the following identities: R ( \ . / I ) = R ( M ) - R ( X ), In other words, R R(Id) = Id . is a contravariant functor from the category of groups and homomorphisms to the category of topological spaces and continuous maps.

And then This yields the following presentation: F W1 Figure 5 These two presentations, of course, are of the same type. considers F as identified with the target of We shall prefer the latter perspective. Abelianizing, we obtain H1(M31Z): h, The first the latter with the source. ,dg): That is, if we consider the pairing: 38 we It Ccanbining Recall follows havethat the immediately these identity: h identities, is an from orientation the we compute: definition preserving of the homeomorphism intersection product from Fthat: to -F.

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