Topological Classification of Multiaxial U(n)-Actions
نویسندگان
چکیده
Since early 1980s, great progress has been made on the classification of finite group actions on the sphere. Deep but indirect connections to representation theory were discovered. The indirectness is reflected by the existence of non-linear similarities between some linearly inequivalent representations [3], via the equivariant signature operator [MR] (see also [HP??] [4]). Whitehead torsion, which was the cornerstone of the classical theory of lens spaces, plays almost no role at all, especially in the presence of fixed points [12, 13]. On the other hand, the action of positive dimensional groups on topological manifolds has been largely left alone, aside from action by the circle. This paper, inspired by the beautiful results of M. Davis and W. C. Hsiang [8] on concordance classes of smooth multiaxial actions on the homotopy sphere, shows that the classification theory in the topological setting is both completely different and quite comprehensible. For the purposes of this introduction, we will assume that G = U(n) acts on M locally smoothly. In other words, every orbit has a neighborhood equivariantly homeomorphic to an open subset of an orthogonal representation of G. We will concentrate on multiaxial actions, which means that the representations are of the form kρn ⊕ j , where ρn is the defining representation of U(n) on C, and is the trivial representation R. While this may allow different choice of k and j at different locations in the manifold, the results
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