c - . ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS KARL...

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Unformatted text preview: . ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN Abstract. Suppose G is a finite cyclic group and M a closed smooth G manifold. In this paper we will show that there is a nonsingular real algebraic G variety X which is equivariantly diffeomorphic to M and all G vector bundles over X are strongly algebraic. 1. Introduction Let G be a compact Lie group. A real algebraic G variety is a G invariant subset x p 1 ( x ) = = p n ( x ) = 0 in a real G module that is the set of common zeros of a finite collection { p 1 ,...,p n } of polynomials [8]. If M is a closed smooth G manifold and X is a nonsingular real algebraic G variety that is equivariantly diffeomorphic to M , then we say that M is algebraically realized and that X is an algebraic model of M . We call X a strongly algebraic model of M , if, in addition, all G vector bundles over X are strongly algebraic. This means that, up to isomorphism, the bundles are classified by entire rational maps. As needed, we will recall definitions and facts from real algebraic transformation groups from [8] and [11]. Existing results lead us to believe Conjecture 1.1. (Compare [11, p. 32].) Let G be a compact Lie group. Then every closed smooth G manifold has a strongly algebraic model. Our principal result confirms the conjecture in a special case. Theorem 1.2. Let G be a finite cyclic group. Then every closed smooth G manifold has a strongly algebraic model. Let us give a brief review of the history. J. Nash [15] posed the algebraic realization problem for closed smooth manifolds, and this problem has an affirmative answer, see Tognoli [19] and Akbulut-King [1], [2]. For cyclic groups G we showed that closed smooth G manifolds are algebraically real- ized, see [9]. See [8] for other results on the equivariant algebraic realization problem. The tangent bundle of a nonsingular real algebraic variety is strongly algebraic. So, instead of algebraically realizing a manifold together with one Date : October 6, 2008. 1991 Mathematics Subject Classification. Primary 14P25, 57S15; Secondary 57S25. Key words and phrases. Algebraic Models, Equivariant Bordism. 1 2 KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN specific bundle, it is natural to try and algebraically realize the manifold with all of its bundles. That this is possible is Conjecture 1.1. Benedetti and Tognoli [3] proved this conjecture in the case where G is the trivial group. Conjecture 1.1 is true if G is a product of an odd order group with a 2torus, see [11, Theorem B]. This implies Theorem 1.2 in the special case where G is a cyclic group whose order is twice an odd number. Conjecture 1.1 is also true if G is a compact Lie group and the action on M is semifree, see [11], or if G is cyclic and the action on the manifold has only one isotropy type, see [12]. Hanson [13] proved Theorem 1.2 for Z 4 manifolds with one vector bundle in addition to the tangent bundle.bundle in addition to the tangent bundle....
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c - . ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS KARL...

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