Abstract: We investigate the equivariant K-theory ring of a compact G-space X, where G is a compact Lie group; trying to relate its algebraic properties to the topological properties of the action. We adopt the idea of Quillen ([6]) and consider the prime ideal spectrum SpecKa(X). The topological properties of the action are detected by the category C(G,X) whose objects are connected components of fixed point sets on X of cyclic subgroups of the group G. In Sects. 2 and 3 we show how the category C(G, X) determines SpecKa(X). It allows us to relate the properties of Spec KG(X ) to those of the category C(G, X). This is done in Sect. 4 where we describe a stratification of the spectrum and prove that its irreducible components are in one-to-one correspondence with the isomorphism classes of maximal objects of the category C(G, X). Our theorems applied when X is a point give a different approach to Segal's results ([8]) on the supports of prime ideals in the representation ring of the group. Such a possibility was pointed out by Quillen in the paper cited above. In Sect. 5 we show that the prime ideal spectrum of the equivariant Ktheory ring determines in a weak sense the category of fixed point sets of cyclic subgroups. More precisely, an equivariant morphism of spaces with group actions induces a bijection of prime ideal spectra of equivariant K-theory if and only if it defines an equivalence of corresponding categories. A similar theorem was proved by Quillen for equivariant Borel cohomology with modp coefficients and categories of fixed point sets of elementary abelian p-subgroups ([6]) but his proof can not be applied directly to equivariant K-theory. In the last section we describe homomorphisms of compact Lie groups inducing equivalence of categories of cyclic subgroups. The resulting theorem together with the results mentioned above gives us a generalization of an elementary theorem which says that for finite groups, any homomorphism inducing an isomorphism of the representation rings must be an isomorphism.
Publication Year: 2005
Publication Date: 2005-01-01
Language: en
Type: article
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Cited By Count: 3
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