An extension of Casson's invariant - download pdf or read online

By Kevin Walker

ISBN-10: 0691025320

ISBN-13: 9780691025322

ISBN-10: 0691087660

ISBN-13: 9780691087665

This e-book 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 distance of conjugacy periods 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 said 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 ordinary facts of the lifestyles of l. it's also proven that after 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

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The associated fibers CP;l' ... ,CP;r are called the jumping lines of the bundle E and are clearly invariants of the isomorphism type of E. Indeed, on the compliment 42 RALPH L. COHEN AND R. JAMES MILGRAM e- of these lines U~ CP;;' E is holomorphically trivial with trivialization uniquely determined by the trivializations on CP~ and the other Cpl, t, [H]. Thus, we can view E as obtained from a trivial bundle C 2 x by gluing in bundles over D x Cpl where D is a small disk neighborhood of the origin in C replacing the original trivial bundle in small neighborhoods of the jumping lines CP;3..

Thus the rational function spaces Ratk (or equivalently the moduli spaces M k ) admit the same homotopy type as the ambient spaces 0%82 (or equivalently Bk ) through a "stable range" of dimensions:::; k - 1. Now the rational function spaces have even more structure. For example we have the following. 8. L : Ratk x Ratr---+Ratk+r so that the following diagram homotopy commutes: 0%82 x 0~82 where u is the loop sum operation. Proof. Let a E Ratk and f3 E Ratr . Since based rational functions are determined by their roots and poles, a and f3 can be viewed as being given HOMOTOPY OF GAUGE THEORETIC MODULI SPACES 31 by configurations of points in the complex plane C each labelled according to whether it is a root or a pole and by a positive integer determining the multiplicity.

C. K. Murray, On the construction of monopoles for the classical groups, Comm. Math. Phys. 122, 35-89. C. Kirwan, On spaces of maps from Riemann surfaces to Gmssmannians and applications to the cohomology of moduli of vector bundles, Ark. Math. 24(2), 221-275. B. Lawson, The theory of gauge fields in four dimensions, CBMS Reg. Conf. Ser. AMS, 1985. M. J. Milgram, Some spaces of holomorphic maps to complex Grossmann manifolds, Jour. Diff. Geom. 33(2), 301-324. M. J. Milgram, On the moduli space of SU(n) monopoles and holomorphic maps to flag manifolds, J.

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