This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonsingular 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 metatheorem 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)....
View
Full
Document
This note was uploaded on 07/26/2009 for the course MATH 215 taught by Professor Heiner during the Spring '09 term at Hawaii.
 Spring '09
 Heiner
 Algebra

Click to edit the document details