Finite Loop Space with Maximal Tori have Finite Weyl Groups by Larry Smith Yale University Mathematisches Institut Mathematics Department Bunsenstrasse 3/5 New Haven, Connecticut D34 G"ottingen USA 06520 West Germany Summary: A finite loop space X is said to have a maximal torus if there* * is a map f : BT -! BX where T is a torus such that: _ rank(T ) = rank(X) _ the homotopy fibre of f has the homotopy type of a finite complex. The Weyl group Wf of f is the set of homotopy classes w : BT -! BT such that w BT -! BT f & . f BX homotopy commutes. In this note we prove that Wf is always finite. Acknowledgement I would like to thank the members of the Yale mathematics department for their hospitality during my stay, and to the Ministieurm f"ur Wi* *ssenschaft und Kultur in Hannover for relieving me of teaching responsibilities during the* * 1990 winter semester. By a finite loop space we understand a topological space X such that _ X has the homotopy type of a finite complex _ X has the homotopy type of a topological group. N.B. It is not assumed both structures may be realized on the same space simult* *aneously. A compact Lie group is a finite loop space, but there are well-known examples (* *see for example the book of Richard Kane [2] and the references there) of finite loop s* *paces which are not Lie groups. An important idea, introduced by D. Rector [7] (see also [6* *; I], [6; II], [6; III] and [5]) for the study of finite loop spaces is that of a maximal toru* *s. DefinitionD:efinitionD:efinitionI:f X is a finite loop space, a maximal tor* *us for X is a map f : BT ! BX where B(-) is the classifying space functor and T is a torus, such that _ rank (T ) = rank(X) _ the homotopy fibre F of f has the homotopy type of a finite complex. A finite loop space need not have a maximal torus as Rector showed [7], and if * *X is a fake Lie group, then it has a maximal torus if and only if it is a Lie group [6; I-I* *II], [5]. For a finite loop space with a maximal torus, Rector adapted a further concept from L* *ie theory, namely the Weyl group. DefinitionD:efinitionD:efinitionI:f X is a finite loop space with a maximal* * torus f : BT ! BX then the Weyl group of f, Wf, is the group of homotopy classes of maps w : BT ! BT such that homotopy commutes. 1 Notice that [BT; BT ] ' GL(n; Z), where n = rank (T ), so Wf GL(n; Z). If G* * is a compact connected Lie group and X a fake Lie group (e.g. X = G) of type G (se* *e [6; I] for definitions) with a maximal torus f : BT ! BX (e.g. f = Baewhere ae : T ,! G is the inclusion of a maximal torus) then Wf * *' WG . However for a general finite loop space there seems no reference for the fact* * that Wf is a finite group.y We rectify this in the following: TheoremTh:eoremTh:eoremLe:t X be a connected finite loop space and suppos* *e X has a maximal torus f : BT ! BX: Then the Weyl group Wf is finite, and |Wf| is a divisor of d = d1 : :d:n w* *here (2d1 - 1; : :;:2dn - 1) is the type of X. The type of a finite loop space X (or H-space) is defined following H. Hopf (* *see for examole [3]) who showed H*(X; |Q) = H*(S2d1-1x : :x:S2dn-1; |Q) where the integers 2d1 - 1; : :;:2dn - 1 are called tye type of X, and n its * *rank. If X is a Lie group, then the rank as just defined coincides with its rank as a Lie gro* *up. ProofP:roofP:roofL:et X have rank n and type (2d1- 1; : :;:2dn - 1). By t* *he Leray-Samelson theorem [3] it follows that for sufficiently large primes p or p = 0 H*(X; IFp) ' E(u1; : :;:un) where (IFp := Z=p; IF0= |Q) degui = 2di- 1 : i = 1; : :;:n: _________________________ y In [8] Rector and Stasheff state that this is the case, but give no proof. It* * is also hard to imagine how they intended to prove this with the methods available at that time. 2 From the Milnor-Moore spectral sequence [4] it then follows H*(BX; IFp) ' IFp[ae1; : :;:aen] where degaei = 2di : i = 1; : :;:n: Let F denote the fibre of the maximal torus f : BT ! BX: Consider the Serre spectral sequence {Er; dr} Er ) H*(BT ; IFp) E2 = H*(BX; IFp) H*(F ; IFp): Since F is finite, E**2is a finitely generated H*(BX; IFp) module. If p = 0, * *or p is sufficiently large, H*(BX ; IFp) is a noetherian ring, and hence an easy argume* *nt shows that H*(BT ; Fp) is a finitely generated H*(BX ; IFp) module. Recall H*(BT ; IFp) ' IFp[t1; : :;:tn] where degti = 2 : i = 1; : :;:n: Therefore f* : H*(BX ; IFp) ! H*(BT ; IFp) must be monic for H*(BT ; IFp) to be finitely generated over H*(BX ; IFp), and * *further- more by MacCauley's theorem [9] H*(BT ; IFp) is a free H*(BX; IFp) module. From this point on assume that p is a prime that is sufficiently large and p 6 * *|d = d1 : :d:n. Then according to Adams-Wilkerson [1] there exists an essentially unique embedd* *ing ' : H*(BX ; IFp) ,! H*(BT ; IFp) 3 and a finite group W (p) GL(n ; IFp) with '(H*(BX ; IFp)) = H*(BT ; IFp)W (p) and |W (p)| = d: The uniqueness of ' allows us to suppose that ' = f*. By the very definition of* * Wf we have H*(BT ; IFp)W(p) H*(BT ; IFp)Wf : Let F F (-) denote the field of fractions functor, and set H*(BT ; IFp) = B. Th* *en F F (B)W(p) = F F (BW(p)) F F (BWf ) F F (B)Wf where the first equality results from [6; I.3.2]. By Galois theory it follows t* *hat the image of Wf under the reduction homomorphism ae : GL(n; Z) ! GL(n; IFp) is contained in W (p). Suppose that |Wf| > d. Let 1 = w0; w1; : :;:wd 2 Wf be d + 1 distinct elements. Choose p even larger if necessary, so that ae p 6 |wr(i; j) - ws(i; j) : 01 r; si;dj n Then the mod p reduction of these elements would have to be distinct, contrary * *to the fact ae(Wf) W (p); |W (p)| = d: ___ Therefore Wf is finite, and p is monic if p is large enough. |__| 4 The preceding theorem answers yet one more question about the Weyl group of a m* *aximal torus for a finite loop space, but leaves open many more. In particular: _ Is the Weyl group of a finite loop space with maximal torus nontrivial? _ Is H*(BX ; IFp) the ring of invariants of Wf acting on H*(BT ; IFp) ? Affirmative answers would classify the possible Weyl groups as the groups gener* *ated by reflections and answer in the affirmative the question of whether the type of a* * finite loop space with maximal torus must coincide with that of a Lie group. 5 References 1. Adams, J. F. and C. W. Wilkerson, Polynomial Algebras over the Steenrod Alge* *bra, Ann. of Math 110 (1980), pp. 96-143. 2. Kane, R. M., Homology of Hopf Spaces, North Holland, 1988. 3. Milnor, J. and J. C. Moore, On the structure of Hopf Algebras, Ann. of Mat* *h 81 (1965), pp. 211-264. 4. J. C. Moore, Algebre Homologique et Homologie des Espaces Classifiant, Semi* *nar Cartan 1959/60 Expose 7. 5. Notbohm, D., Maps Between Classifying Spaces and Applications, Mathematica G* *"ottingensis Heft 20 (1991). 5. Notbohm, D., and L. Smith, Fake Lie Groups and Maximal Tori I, II, III Math.* * Ann. 288 (1990) 637 - 661, (to appear). 6. Rector, D., Subgroups of Finite Dimensional Topological Groups, JPPA 1 (1971* *) pp. 253-273. 7. Rector, D. and J.D. Stasheff, Lie Groups from a Homotopy Point of View, in L* *ocalzation in group : :S:pringer LNM 418 (1974), pp. 121 - 131. 8. Zariski, O. and P. Samuel, Commutative Algebra, Vol. II (Appendix 7). 6