Homotopy fixed point methods for Lie groups and finite loop spaces W.G. Dwyer and C.W. Wilkerson University of Notre Dame Purdue University x1. Introduction A loop space X is by definition a triple (X; BX; e) in which X is a space, BX is a connected pointed space, and e:X! BX is a homo- topy equivalence from Xto the space BX of based loops in BX . We will say that a loop space X is finite if theintegral homology H (X; Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem. 1.1 Theorem. If X is a finite loop space, then forany prime number p the cohomology algebra H (BX; Fp) is finitely generated (as an algebra). Any compact Lie group G is a finite loop space: G is a loop space because it is homotopy equivalent to BG (where BG is the usual clas- sifying space for principal G-bundles) and G is finite because it is a compact smooth manifold. It was proved in [22] that for a compact Lie group G the algebra H (BG; Fp) is finitely generated. Theorem 1.1 answers!an old question of J. Moore by extending this result to finite loop!spaces. In fact, we obtain a new homotopy theoretic proof of the result!in![22]. ! Themethod b ehind the proof of Theorem 1.1 is really more interesting than!the!theorem itself. We immediately (x2) reduce 1.1 to an analogous theorem!about!other loop spaces called p-compact groups, andthen we study!p-compact groups in great detail. Bydefinition a p-compact group is!purely!homotopy theoretic in nature: it is essentially just a finite loop space!with all of its structure concentrated densely at the single prime p. Our!experience!shows, however, that a p-compact group possesses much of!the rich internal structure of a compact Lie group. It seems to us that p-compact!! groups are remarkable objects, and one of the main goalsof the!paper!is to introduce them and study theirprop erties. ! The authors were supported in part by the National Science Foundation. The first author would like to thank theMIT mathematics department for its hospitality during part of the time this work was carried out. 2 Homotopy fixed point methods We begin the paper by showing that the elementary geometricap- paratus of Lie group theory can be re-interpreted so as to apply to a p-compact group X: this includes the notion of a subgroup of X, cen- tralizer of a subgroup of X, quotient of X by a central subgroup, etc. The homotopy theoretic definitionsof some of these concepts are given in x3, and we spend the next few sections showing that the definitions are plausible ones. Many unusual questions come up (what is an element of order p in X? are there any? is the centralizer in X ofan element of order p a subgroup of X? is an element of order p contained in its centralizer?) andnew techniques have to be devised to answer them. However, when the dust has settled and the machinery is in place, we are able to use a straightforward inductive scheme (cf. 1.2) to produce a maximal torus T in a p-compact group X, show that the normalizer N(T ) of T is an extension of T by a finite pseudoreflection group W (the Weyl group of X), andprove that the homogeneous space X=Np(T) has Euler characteristic prime to p. (Here Np (T) is the inverse image in N(T ) of a p-Sylow subgroup of W .) It is not hard to see directly that H (BNp (T);Fp) is a finitely generated algebra, and finite generation for H (BX; Fp) follows from a standard lemma that uses the Becker- Gottlieb transfer of the fibration X=Np(T )! BNp(T )! BX. Thephilosophy behind this paper is due to Rector [20]; in drawing a serious analogy between the homotopy theory of loop spaces and the geometry of Lie groups we are following his lead. From a practical point of view we are heavilydep endent on the results of Lannes [15]; in combination with our previous work [9], these results allow us to study homotopy fixed point sets of finitep-group actions with the same Smith theory techniques which are used in the classical case to study fixed point sets (x2). Some of the homotopy fixed point sets which arise are described below (1.3). 1.2 A sketch of the main argument. Our study of maximaltori in p-compact groups is directly parallel to an unortho dox geometric method for studying maximal tori in a compact Lie group. We will describe this Lie theoretic approach and indicatewhere in the paper the corresponding homotopy theoretic points come up.Let G be a compact Lie group. A toral subgroup T of G is a subgroup isomorphic to a pro ductSO (2)n of circle groups (6.3); T is called a maximal torus (8.9) if it is equal to the identity component of its centralizer (3.4) in G. If G is a discrete group, then the identity subgroup isa maximal torus for G. To construct such a maximal torus if G is not discrete, we first find a nontrivial element of order p in the identity comp onent of G (5.4), and then show that W. Dwyer and C. Wilkerson 3 f1 : Z=p1 ! G (5.5). The closure A of the image off1 (6.6, 6.7) is a nontrivial connected abelian subgroup ofG (connected because Z=p1 has no finite quotients) and so isa toral subgroup of G. Let H be the centralizer in G of A and let T=A be a maximal torusin the quotient group H=A; such a T =A can be assumed to exist by induction, since H=A is a group of smaller dimension (6.13) than G. Let T be the inverse image of T=A in H. It is not hard to see that T is a maximal torus for H (pro of of 8.13) and consequently (8.14) amaximal torus for G (observe in this last connection that T contains A (8.2, 8.11) and so the centralizer of T in G is equal to the centralizerof T inside the centralizer of A in G (8.4), i.e., equal to the centralizer of T in H). Now let N (T) ae G be the normalizer of T in G (9.8); since T is a maximal torus, N(T )=T is a finite (discrete) group (9.5). By inspection the fixed point set of the left translation action of T on the homogeneous space G=T is exactly the set N(T )=T , and so by elementary transformation group theory the Euler characteristic of G=T is equal to the cardinality ofN T=T (9.5). We are using here the fact that under reasonable hypotheses the Euler characteristic (4.3) of the fixed point set of a torus action on a compact space Y is equal to the Euler characteristic of Y (4.7). It follows from the multiplicativity of the Eulercharacteristic in finite coverings (4.14) that the Euler characteristic of G=N T is 1. A similar argument shows that any other maximal torus T0of G is conjugate to T (9.4) (the action of T0 on G=T must have a fixed point (8.11) because the Euler characteristic of G=T is nonzero) and that if G is connected then T is actually equal to its own centralizer in G (9.1). It is now possible to prove that H (B G;Fp) is a finitely generated algebra byusing transfer (9.13) in the fibration G=N (T )! BN (T )! B G(2.4) and appealing to the arguments of x12 for the fact that H (BN (T);Fp) is finitely generated. 1.3 Homotopy fixed point sets. A fundamental part of our technique involves interpreting group theoretic constructions as fixed point sets, identifying the corresponding homotopy fixed point sets, and then using the homotopy objects as substitutes for the geometric ones. Here are some examples. (1) The centralizer in G of the image of a homomorphism f: K! G is the fixed point set of the action of K on G via f by conjuga- tion. The correspondinghomotopy fixed point set is the space of sectionsover BK of the fibration with fibre G associated to this conjugation action of K on G. Up to homotopy this fibration is the pullback over Bf of the freelo opspace fibration over BG [11];its space of sections is the loop space on the component of