An extension of Casson's invariant by Kevin Walker

By Kevin Walker

This publication 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. allow R(X) denote the distance of conjugacy periods 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 effectively 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 less than Dehn surgical procedure is proved. The formulation contains Alexander polynomials and Dedekind sums, and will be used to provide a slightly ordinary evidence 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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10), Xf,XE 3(U) . 19) If df is expressed in terms of coordinates by df = h i dx1 + ... + h"dx", then (2. 11) and (2. +af dx". , n - 1, which agrees with (2. 22) where deg w denotes the degree of co, and where f A T1 means the same as frl if f E S2° U. This mapping d is called the exterior derivative. The proof of the theorem, which depends on nothing more than combining the rules of multivariable calculus with those of exterior algebra, will be given a little later. Condition (2. 22) is tantamount to the equality of the mixed second partial derivatives, and could be called the Iteration Rule.

V) w in L (V -4 W) , and extend to all of V` 0 W by linearity. The tensor product of 1-forms co', w2, denoted co' 0 0)2, is the map which assigns to each y e U the bilinear map w' ®w2 (y) E (TYR") * ® (TYR") * . w' ®w2 (y) (4, c) = co' (y) (4) (02 (y) (c). Thus for vector fields X, Y, we may write: w' ®w2 (X, y) = (co' . X) (w2 - Y) E C`° (U). For example, dx ® xdy (eza xe;. The relationship with the exterior -, ay + yaz) = product is: w1 AO)2 = w1 ®0)2-w20 w1. i where the { h13 } are smooth functions on U.

L f g. 2. , without using the notion of a derivation), that for any vector fields X and Yon an open set U c R", [X. Y1 defined by (2. 5) is indeed a vector field. , x"). Show that the two second-derivative terms cancel out. 3. Verify the Jacobi identity (2. 6) for vector fields. Hint: Work in terms of derivations; don't differentiate anything! 4. Show that if X and Y are the vector fields on U = R3\ { 0} given below, then LXY = 0: X= -{x2+y2+z2}-3/2{xa +ya +za ax ay az }; Y = -ya +xa . -3 Hint: For brevity, take r = {x2 + y2 + z2 } 1 /2 and note that (xar y ax ay -3 y ax ) = 0.

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