An extension of Casson's invariant by Kevin Walker

By Kevin Walker

This e-book describes an invariant, l, of orientated rational homology 3-spheres that is 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). permit (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is said to be an thoroughly outlined intersection variety of R(W) and R(W) inside of R(F). The definition of this intersection quantity is a fragile job, because the areas concerned have singularities. A formulation describing how l transforms less than Dehn surgical procedure is proved. The formulation includes Alexander polynomials and Dedekind sums, and will be used to provide a slightly uncomplicated facts of the lifestyles 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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Extra info for An extension of Casson's invariant

Example text

If p is a periodic point of Ot with period T and {p} is a closed set, then OT (p) = p. Proof. Let A =IT > 0 1 0T(p) = p}. By the definition of period, T = inf A. So, T E cl(A). Because {p} is closed, (T, p) E cl(A) x {p} = cl(A x {p}). 2. Periodic Points 29 Since 0 is continuous and {p} is closed, OT (p) = O(T, p) E cl(q(A x {p})) = cl({p}) = {p}. Therefore, OT (p) = p. 5. Let qt be a flow on a topological space X. If p E X and O' (p) = p for some real number T, then OnT (p) = p for every integer n.

So, each orbit lies within a level set of F. To determine the Poincare recurrent set of Ot we analyze the level sets of F. Case I. F(x, y, z) = 0. In this case the level set of F is the z-axis. Along the z-axis, x = y = 0, and the system of differential equations reduces to the constant differential equation z=1. Thus, the z-axis is an orbit of Ot, and no point on the z-axis is Poincare recurrent. Case II. F(x, y, z) = 2 for r > 0. r 2. Recurrent Points 54 Parametrize the level set F(x, y, z) = 1/r2 by x(a, 8) _ r2 (r -} - 1 cos a) cos 0, = (r + r2 - 1 cos a) sin 9, r2 - 1 sin a, z(a, 9) = y(a, 9) where a e S' and 9 E S1.

The fixed set of a flow on a Hausdorff topological space is closed. Proof. Let Ot be a flow on a Hausdorff topological space X. We shall prove that X \ Fix(gt) is open. Let x E X \ Fix(gt). Thus, there exists a real number T such that 0'(x) L x. Since X is Hausdorff, there exist disjoint open subsets U and V of X such that 0'(x) E U and x E V. Let A = 0--r(U) n V. Since 0' is continuous, A is an open set. We shall show that A C X \ Fix(ot). If y c A, then 0'(y) E U and y E V. Because U and V are disjoint, qT(y) y.

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