By Kevin Walker

This ebook describes an invariant, l, of orientated rational homology 3-spheres that's a generalization of labor of Andrew Casson within the integer homology sphere case. permit R(X) denote the gap of conjugacy sessions of representations of p(X) into SU(2). enable (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is asserted to be an thoroughly outlined intersection variety of R(W) and R(W) within R(F). The definition of this intersection quantity is a fragile activity, because the areas concerned have singularities. A formulation describing how l transforms below Dehn surgical procedure is proved. The formulation contains Alexander polynomials and Dedekind sums, and will be used to provide a slightly user-friendly evidence of the life of l. it's also proven that once M is a Z-homology sphere, l(M) determines the Rochlin invariant of M

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**Example text**

The answer is affirmative. We start with an example of an infinitely generated group with such a property. 1. 1/ . 1/ D 1: Indeed, note that for every n 2 N there is an inclusion jn W Zn ! 1/ . 1/ . 1/ cannot be finite. To construct finitely generated groups with infinite asymptotic dimension, we introduce the notion of a wreath product. Let G and H be finitely generated groups. H I G/ of finitely supported functions f W H ! G. This set has a natural group structure L through coordinate-wise multiplication and as a group can be identified with .

This implies that h is in the center of H . Hence the kernel of the adjoint representation is abelian. The group A, as a locally compact abelian Lie group, has finite asymptotic dimension. n; R/ is the Lie group of all orthogonal matrices. n; R/ is solvable; we leave this fact as an exercise. n; R/ has finite asymptotic dimension, by the following lemma. 5. If G is an almost connected solvable Lie group, then asdim G < 1. Ni =Ni 1 / < 1. 6. Let G be an almost connected Lie group. Then G has finite asymptotic dimension.

8. Let X be a bounded geometry metric space with finite asymptotic dimension. Then X has finite decomposition complexity. The proof requires a theorem of Dranishnikov [70], see also [78], stating that a bounded geometry metric space embeds coarsely into a Cartesian product of finitely many trees. Since trees have asymptotic dimension 1, they also have finite decomposition complexity. 5 we conclude that a product of finitely many trees also has finite decomposition complexity and, consequently, any space that coarsely embeds into such a product.