ALGEBRAIC REALIZATION FOR SIMPLE GROUPS KARL HEINZ DOVERMANN AND VINCENT GIAMBALVO Abstract. Let G be a finite simple group with elementary abelian Sylow 2 subgroup. We show that every closed smooth G manifold has a strongly algebraic model. 1. Introduction Let G be a compact Lie group. If M is a closed smooth G manifold and X is a nonsingular real algebraic G variety (for definitions see Section 8 or [16]) 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 the bundles are classified, up to equivariant homotopy, by entire rational maps to equivariant Grassmannians with their canonical algebraic structure (see Section 8.2). Exisiting results motivate Conjecture 1.1. [18, p. 32] Let G be a compact Lie group. Then every closed smooth G manifold has a strongly algebraic model. In support of the conjecture we will show: Theorem 1.2. Let G be a finite simple group with elementary abelian Sylow 2 subgroup. Then every closed smooth G manifold has a strongly algebraic model. We proved Theorem 1.2 for A5 “ PSLp2; 5q in [13] and for PSLp2; qq where q is a power of 2, or q “ pn where p is a prime congruent to 3 or 5 modulo 8 and n is odd in [14]. The remaining finite simple groups with elementary n abelian Sylow 2 subgroup are the Janko group J1 and the Ree groups Rp3 q for n ¡ 1 odd, see [38], and we treat them in this paper. In Section 5 we verify the conjecture for Rp3q as well, even though the group is not simple. For the most part we use the major structure theorems for these simple groups and combine them with results from real algebraic transformation groups, though some of the latter need to be upgraded. Theorem 1.3. Every closed smooth D2p1 ˆ¨ ¨ ¨ˆD2pk manifold, for p1; : : : ; pk not divisible by 4, has a strongly algebraic model. Date: August 13, 2020. 1991 Mathematics Subject Classification. Primary 14P25, 57S15; Secondary 57S25. Key words and phrases. Algebraic Models, Equivariant Bordism. 1 2 KARL HEINZ DOVERMANN AND VINCENT GIAMBALVO This theorem provides a step in the proof of Theorem 1.2. It is a gener- alization of the main result of [12]. For the proof see Section 10. Another result that we need is Theorem 1.4. Let N be a compact Lie group and H a closed normal sub- group. Let X be a closed smooth N manifold so that for all x P X the isotropy group Nx at x is equal to H. Assume that every representation of H is the restriction of a representation of N. Then X has a strongly algebraic model. Special cases of the last result, with and without the assumption on the representations, are known [18, 36, 19, 12]. We just need a little more than what is in the literature. Our proof, see Section 9, makes use of results in [9, 10, 8]. 1.1. A brief history. Nash [29] asked whether every closed smooth mani- fold has an algebraic model. This was confirmed by Tognoli [37]. Benedetti and Tognoli [4] showed that every closed smooth manifold has a strongly algebraic model. See also an article by Ivanov [25] and the work of Akbulut and King [1, 2]. Various authors developed a body of work that allows us to study strong algebraic realization questions in the equivariant category and we have been able to verify Conjecture 1.1 under certain assumptions and for some families of groups. The conjecture holds if the action is semifree and if the group is the product of an odd order group and a 2 torus [18, Theorem B]. The conjecture holds when G is cyclic, see [19], [15], and [20]. Results that we need for the current paper are collected in Section 7. One may hope that the (strong) algebraic realization problem for a group G reduces to one of its Sylow 2 subgroup. It did so in case the Sylow 2 subgroup of G is cyclic, and Conjecture 1.1 holds for finite groups with cyclic Sylow 2 subgroups, see [13]. Conjecture 1.1 also holds for elementary abelian 2 groups, so one may attempt to prove Conjecture 1.1 for simple finite groups whose Sylow 2 subgroup is elementary abelian. After [14] this paper is the second and final part of this project. 1.2. Organization of the paper. In Section 2 we establish some basic notation. Proposition 2.4 embodies our basic strategy to solve the realization problems. Its proof is postponed until Section 7. Because of its importance throughout, we also review the process of blowing up a manifold along a submanifold. In Section 3 we prove Theorem 1.2 for the group J1. In Section 4 we we summarize basic knowledge about the Ree groups that we use. In Section 5 we verify Conjecture 1.1 for the Ree group Rp3q. The discussion serves as a road map for the proof of Theorem 1.2 for the simple Ree groups in Section 6. Assuming certain topological results, the proof is combinatorial and group theoretic. In Section 7 we fill in topological results used earlier. Many are quoted from the literature, some are new and proved here. Section 8 is a service to readers who are less familiar with the topic of this paper. There we provide definitions and results that are familiar August 13, 2020, ALGEBRAIC REALIZATION FOR SIMPLE GROUPS 3 to many, but not all. In Section 9 and 10 we catch up with the proofs of Theorems 1.3 and 1.4. 1.3. Notation. We use the letters C, D, M, A, and S to denote cyclic, dihedral, metacyclic, alternating, and symmetric groups. Given groups L and H and a homomorphism ' : H Ñ AutpLq we have the semidirect product of L and H. We denote it by L ¸ H. There is a short exact sequence t1u ÝÝÝÝÑ L ÝÝÝÝÑ L ¸ H ÝÝÝÝÑ H ÝÝÝÝÑt1u: Typically the homomorphism ' will be suppressed in the notation because it is either understood or not relevant. We may write D2q for the dihedral group of order 2q instead of Cq ¸ C2 and D6 for S3. The metacyclic group M21 would be C7 ¸ C3. In Section 11 we separately prove one result that we use in the proof of 2 Theorem 1.2 for the Ree groups. Let N “ ppC2q ˆ D14q ¸ C3, where C3 2 acts on both factors, acting without fixed points on pC2q and on the cyclic subgroup C7 Ă D14. Theorem 1.5. Closed smooth N manifolds have strongly algebraic models. From now on we assume that our groups are finite. It makes some dis- cussions easier. We also allow manifolds to have components of different dimensions. That simplifies our notation and language. 1.4. Thanks. We like to thank Ron Solomon for his generous help with the group theory. Critical questions of Mikiya Masuda encouraged us to clean up some arguments. 2. Preparation, strategy, and blow{ups Let G be a group. The set of subgroups of G is partially ordered. We follow the convention that K ¤ H if K Ď H. According to this convention t1u is minimal and G is maximal. If H is a subgroup of G then we denote its conjugacy class by pHq. The partial order on the set of subgroups of G induces a partial order on the set of conjugacy classes. Suppose M is a closed smooth G manifold. Given x in M, its isotropy ´1 group is Gx “ tg P G | gx “ xu. Then Ggx “ gGxg . For H Ď G we set H pHq 1 (2.1) M “ tx P M | Gx Ě Hu & M “ tx P M | Gx Ě H P pHqu: A point x P M belongs to M H if it is left fixed by all elements of H, and x belongs to M pHq if it is left fixed by all elements of a group H1 that is H pHq conjugate to H. The H fixed point set M is an NGH manifold and M is G invariant. Notation 2.1. Let M be a closed smooth G manifold. (1) If A is a G invariant submanifold, then BpM; Aq denotes the blow{up of M along A. For details of the construction, see Section 2.1. 4 KARL HEINZ DOVERMANN AND VINCENT GIAMBALVO (2) Suppose H a subgroup of G. Denote by RH the union of all com- ponents of M H that are of codimension zero in M, and by RpHq the union of all components of M H1 that are of codimension zero in M, where H1 ranges over all H1 P pHq. The components of RH are also components of M. (3) With H understood, we set M “ MzRpHq. Definition 2.2. Let Y be a closed smooth G manifold and H a nontrivial subgroup of G. We say that we can successfully remove the H fixed point set if pHq (1) The blow{up Y “ B Y; Y exists. (2) Y H “H and [Y L “ H´ ùñ Y¯L “H] for all subgroups L of G. p (3) Y having a strongly algebraic model implies that Y has a strongly algebraicp model. p p The point of Definition 2.2 is that a successful blow{up reduces the strongly algebraic realization problem for Y to the one of Y , and the lat- ter problem may be easier to answer because Y L is nonempty for fewer subgroups of G than for Y .