# Casson's Invariant for Oriented Homology Three-Spheres: An by Selman Akbulut

By Selman Akbulut

In the spring of 1985, A. Casson introduced an attractive invariant of homology 3-spheres through structures on illustration areas. This invariant generalizes the Rohlin invariant and provides fantastic corollaries in low-dimensional topology. within the fall of that very same yr, Selman Akbulut and John McCarthy held a seminar in this invariant. those notes grew out of that seminar. The authors have attempted to stay with reference to Casson's unique define and continue via giving wanted info, together with an exposition of Newstead's effects. they've got usually selected classical concrete methods over normal tools. for instance, they didn't try and supply gauge conception causes for the result of Newstead; in its place they his unique techniques.

Originally released in 1990.

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2 that the ring of coefficients for one version of elliptic cohomology may be identified with where and are modular forms of weights 2 and 4, respectively, and the invariance subgroup is contained in Formally and allowing other invariance subgroups we have the following definitions. DEFINITION. Let and be of finite index. A function f : from the upper-half plane to the complex numbers is called a modular form of weight k with invariance subgroup if: 1. f is holomorphic on 2. for all 3. for all and has a Fourier expansion in nonnegative powers of and some suitable natural number N.

5. 2. 1. 2. is an isomorphism for all finite groups G if and only if is an isomorphism when G is abelian. 3. 3. Let be a multiplicative, complex-oriented cohomology theory and G a finite group. Then the natural map: induced by restriction is an isomorphism. Work of M. Hopkins, N. Kuhn, and D. 2) that and the construction of is natural with respect to restriction. The category suffices because we suppose that is free-abelian and we are concerned with quotients of in G. 1, it suffices to assume that The space of class functions: is likewise free with the same rank Hence we must show that the matrix representing is nonsingular.

An map f : M genus with point is regarded as a fibration. 5. A graded odd formal group law F over where the constant given by: That is, we take the Chern class of the tensor product of two line bundles. To say that F is odd and graded means that and F(–x, –y) = –F(x,y). Assuming that we know the universal property of the formal group law in cobordism Cases 4 and 5 are equivalent (compare the discussion in Chapter 1). There is a close connection, explained for example in [96, Chap. 4], between oriented cohomology theories over and anticommutative graded rings R equipped with a formal group law F.