o - ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ONE...

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Unformatted text preview: ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ONE ISOTROPY TYPE KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN Abstract. Suppose G is a cyclic group and M a closed smooth G manifold with exactly one isotropy type. We will show that there is a nonsingular real algebraic G variety X which is equivariantly diffeomor- phic to M and all G vector bundles over X are strongly algebraic. 1. Introduction A vector bundle over a variety is said to be strongly algebraic if it is classified, up to homotopy, by an entire rational map to a Grassmannian with its canonical algebraic structure. We will catch up with definitions and background material in Section 3. Theorem 1.1. Suppose G is a cyclic group and M a closed smooth G manifold with exactly one isotropy type. Then 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. This theorem is a key ingredient in the proof of Theorem 1.2. [15] Suppose G is a cyclic group and M a closed smooth G manifold. Then 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. Our work is motivated by the Nash conjecture [23], which was shown to hold in [29], [1], and [2]. Early results on algebraic realization are due to Seifert [24]. Adding a smooth action and vector bundle theory as structure on the manifold, one has the more general Conjecture 1.3. (see [12] and [15]) Let G be a compact Lie group and M a closed smooth G manifold. Then 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. Previously, the conjecture has been proven in the following special cases: (1) G is the trivial group, see [5]. Date : May 19, 2009. 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 (2) G is the product of an odd order group and a 2torus, see [12]. (3) The action of G on M is semifree, see [12]. (4) All points of M have orbit type H , and NH/H is of odd order, see [28]. J. Hanson showed in [17] that for a closed smooth Z 4 manifold and one Z 4 bundle over M one can find a non-singular real algebraic Z 4 variety X , so that the diffeomorphism pulls back to a strongly algebraic bundle over X . Let us sketch the bigger picture. Tognols important breakthrough in the discussion of the algebraic realization problem was its reduction to a bordism problem. The following meta-theorem is well supported: General Principle. [19, p. 154] If a topological situation is cobordant to an algebraic situation, then it is isomorphic to an algebraic situation (if you have the right notion of bordism and do a lot of work)....
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This note was uploaded on 07/26/2009 for the course MATH 215 taught by Professor Heiner during the Spring '09 term at Hawaii.

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o - ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ONE...

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