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Additional info for Casson’s invariant for oriented homology 3-spheres : an exposition.
Hence, every closed model. closed 3-manifold has an Heegard 3-manifold is homeomorphic to an Heegard We shall work with Heegard models for the most part. This is primarily done for purely computational and technical reasons. »bg holes with 3 discs. ,h~l(bg) 2 M3 F and by attaching discs to sphere to get and capping to attach Wt Wj. 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.
2, it follows that the conjugation action descends to a welldefined faithful action of on Convention: (1) We consider all of the above isomorphisms as identifications. We observe that the natural involution of defined in the previous section, under the above identification with the unit quaternions, is given by the rule: 11 Note: We shall primarily work with have these alternative models. one model than another. 1, we conclude that from Proposition Hence, 12 On the other hand, 2. The space of representations of (a) the functor Let G G.
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. (b) The action of By our SO, previous on R(G) remarks, descends to a faithful action of the conjugation action of SO 3 # : SO 3 χ S 3 Given a group of SU 2 (C)): G, we may define on S3 on S3 S3: > S3 an action of (faithful) . SO 3 on R(G) (or an action This is a natural action in the following sense.