ON THE HOMOTOPY TYPE OF THE CLASSIFYING SPACE OF THE EXCEPTIONAL LIE GROUP OF RANK 4 ALE~S VAVPETI~C & ANTONIO VIRUEL Abstract.Previous work of several authors shows that the exceptional Lie group of rank 4, F4, as a p-compact group, is determined up to isomorphi* *sm by the isomorphism type of its maximal torus normalizer for p > 2. This * *paper considers the case p = 2 proving that F4as 2-compact group is also deter* *mined up to isomorphism by the isomorphism type of its maximal torus normalize* *r. This allow the authors to determine the integral homotopy type of F4 amo* *ng connected finite loop spaces with maximal tori. 1.Introduction One of the major problems in Homotopy Theory is the understanding and clas- sification of finite loop spaces. A loop space L := (L, BL, e) consists of a pa* *ir of spaces L and BL, BL pointed, and a homotopy equivalence e : BL ' L defining a loop structure on L. The space BL is called the classifying space of L. A loop space L := (L, BL, e) is called finite (resp. Fp-finite) if H*(L; Z) (resp. H*(* *L; Fp)) is finitely generated as graded abelian group. Examples of finite loop spaces are * *given by compact Lie groups; for every compact Lie group G, being BG its honest clas- sifying space, there exists a canonical equivalence e : G ' BG which establish* *es a finite loop space structure (G, BG, e) on G. A great inroad in the subject was the advent of p-compact groups. In the cel- ebrated paper [14], Dwyer and Wilkerson introduced the concept of p-compact group, a homotopy theoretic generalization of compact Lie group. Given a prime number p, a loop space X := (X, BX, e) is said to be a p-compact group if X is Fp-finite, and BX is p-complete in the sense of Bousfield-Kan [7]. Again exam- ples of p-compact groups are given by the p-completion of compact Lie groups, t* *he triple G^p:= (G^p, BG^p, e), is a p-compact group when ß0G is a finite p-group.* * In this way p-compact tori appear: a p-compact torus of rank n is a triple (T, BT,* * e) where BT ' K (Zbp) n, 2) is an Eilenberg-MacLane space of degree 2, that is, a p-compact torus of rank n is the p-completion of a rank n torus. Further exam- ples are given by the realization of polynomial algebras, i.e., by pairs ( BX, * *BX), where BX has polynomial mod p cohomology (see [1], [8], [13], [28], [31] and [3* *9]). These purely homotopy theoretic objects, possess much of the rich internal s* *truc- ture of compact Lie groups so it is possible to set up a dictionary translating* * con- structions and arguments from the algebraic theory of groups to the homotopical setting of p-compact groups (see the original [14], or the reviews [11], [20] a* *nd [27]). In particular, it is possible to define: ____________ The first author is partially supported by Ministry of Science and Technology* * of the Republic of Slovenia research grant No. J1-0885-0101-98. The second author is partially * *supported by the DGES grant PB97-1095 and by a Junta de Andaluc'``a Grant FQM-0213. 1 2 ALE~S VAVPETI~C & ANTONIO VIRUEL o Homomorphisms [14, x3.1]: A homomorphism X ____f-Y of p-compact groups is a pointed map BX ____Bf-BY . The homomorphism f is an iso- morphism if Bf is a homotopy equivalence. It is a monomorphism if the homotopy fiber Y=X of Bf is Fp-finite or equivalently if H*(BX, Fp) is a finitely generated module over H*(BY, Fp) via Bf*. o Centralizers [14, x3.4]: For a homomorphism Y _____f-X of p-compact groups, the centralizer CX f(Y ) is defined by the equation BCX f(Y )* * := map (BY, BX)Bf. o Maximal tori [14, Definition 8.9]: A monomorphism T ! X of a p-compact torus into a p-compact group X is a maximal torus if CX (T ) is a p-comp* *act toral group and if CX (T )=T is homotopically discrete. Every p compact group admits maximal tori [14, Theorem 8.13]). o Weyl group [14, Definition 9.2]: Let BTX ____BfT-BX be a maximal torus of a p-compact group X. Assume that BfT is already a fibration and treat WX as the space of self-maps of BTX over BX. Composition gives WX the structure of an associative topological monoid. It is shown [14, Proposi* *tion 9.5]) that WX is homotopically discrete and therefore WX := ß0WX is a (finite) group. Moreover, the action of WX on BTX induces a faithful representation WX- ___- GL (H*QpBTX ) ~=GLn(Q^p) whose image is generated by pseudo reflections, i.e. WX is a pseudo refl* *ec- tion group [14, Theorem 9.7]. o Maximal torus normalizers [14, Definition 9.8]): Let BTX ____BfT-BX be a maximal torus of a p-compact group X. The normalizer of TX denoted by NTX , or simply by NX , is the loop space such that BNTX is the Borel construction of the action of WX on BTX . It has been conjectured (see [23]) that two p-compact groups are isomorphic * *if and only if their maximal torus normalizers are isomorphic, that is, if X and Y* * are p-compact groups such that NX and NY are homotopy equivalent loop spaces, then X ~=Y as p-compact groups. Those p-compact groups that verify the conjecture are called N-determined. It has been shown that for p > 2, all p-compact groups are N-determined (see [2]), but for p = 2, the conjecture fails as maximal torus normalizer cannot co* *ntrol connectivity. So SO(3)^2is not N-determined since its maximal torus normalizer * *is isomorphic to O(2)^2, which is again a 2-compact group, but SO(3)^2and O(2)^2are not isomorphic 2-compact groups, although they have isomorphic maximal torus normalizers. So one is forced to consider a weaker version of the normalizer conjecture f* *or p = 2 (see [29]): we shall say that a 2-compact group X is weakly N-determined if for any 2-compact group Y , X ~=Y as 2-compact groups if and only if there exists an homotopy equivalence of loop space NX ' NY inducing an isomorphism ß0X ~=ß0Y . Of course, N-determined 2-compact groups are weakly N-determined. It has been shown that the 2-compact groups (G2)^2([34]) and O(n)^2([29]) are weakly N-determined 2-compact groups (although they are not N-determined), and that unitary groups ([25]) and DI(4) ([30]) are N-determined . HOMOTOPY UNIQUENESS OF BF4 3 All those previous cases cited above share a common property: the classify- ing spaces realize polynomial algebras which are the invariants of a precise ac* *tion on the cohomology of an elementary abelian 2-group (with a suitable Frobenius twist). There are two more 2-compact group families with that nice property: the 2-completion of simplectic groups Sp(n), and the 2-completion of the exceptional Lie group of rank 4, F4 (see Proposition 2.5). This fact makes centralizer calc* *ula- tions easier, and a tricky use of the 2-stubborn subgroup decomposition ([18]) * *of F4 allows us to prove that the 2-compact group (F4)^2is N-determined, that is, Theorem 1.1. Let X be a 2-compact group with maximal torus normalizer N ___j-X, if N is homotopy equivalent to N(F4)^2as a loop space, then X and (F4)^2are iso- morphic 2-compact groups. The case of the 2-compact groups obtained by the 2-completion of Sp(n) is considered in [33]. The normalizer conjecture can also be stated for finite loop spaces as a weak version of Wilkerson's conjecture on finite loop spaces with maximal tori (see * *[36]): Let L be a connected finite loop space with maximal torus normalizer isomorphic* * to that of a compact connected Lie group G, then it is conjecture that BL is homot* *opy equivalent to BG. If L is a connected finite loop space with maximal torus normalizer N, isomo* *rphic to that of a compact connected Lie group G, proving that BL ' BG is equivalent to prove that BL and BG lie in the same adic genus (see [26]). The rational case is trivial so we fix our attention to the p-completion of BL. As L is finite a* *nd connected, its p-completion gives rise to a p-compact group L^psuch that o its classifying space is BL^p o its maximal torus normalizer is just the fiberwise completion of N by the fibration BT ____- BN ____- BWL = BWG . In other words, p-completion of L, or BL, gives rise to a connected p-compact g* *roup whose maximal torus normalizer is isomorphic to that of the p-compact group G^p. Therefore BL and BG lie in the same genus if and only if the p-compact group G^p is weakly N-determined for all primes p. Previous works show that the group F4 considered as a p-compact group is N- determined for p > 2 (see [25] for p > 3, [35] for p = 3 and [2] for the genera* *l case). Those results combined with Theorem 1.1 allow us to prove: Corollary 1.2. Let L be a connected finite loop space with maximal torus normal- izer isomorphic to that of F4. Then BL is homotopy equivalent to BF4. Organization of the paper: In Section x2 we describe the mod 2 cohomology of BF4. This description allow us to easily calculate the Quillen category of F* *4 at p = 2 in Section x3. In Section x4, we exhibit an interesting connection between the p-stubborn category and the Quilllen category of a Lie group G. Finally, in Section x5 we prove Theorem 1.1 Notation: Here A2 is the mod 2 Steenrod algebra, all spaces are assumed to have the homotopy type of CW-complexes. Given a space Y , we write H*Y for H*(Y ; F2), H*Q^2(Y ) for H*(Y ; Z^2) Q and Yp^ for the Bousfield-Kan (Z=p)1 - completion, or p-completion, of the space Y ([7]). Given a p-compact group X, Qp(X) denotes the Quillen category of X at the prime p (see Section 3). For a compact Lie group G, and a prime p, we write Rp(G) for the p-stubborn category 4 ALE~S VAVPETI~C & ANTONIO VIRUEL (see Section 4). For a group G and an element g 2 G, cg denotes the inner group automorphism induced by conjugation by g. Given two groups K and G, we denote by Mono(K, G) the set of G-conjugacy classes of monomorphisms K-___f-G. Acknowledgements: The authors would like to thank C. Broto and C. Wilkerson for many interesting suggestions. They also would like to thank Centre de Recer* *ca Matem`atica at Bellaterra, for providing the opportunity of meeting and hospita* *lity while part of this work was being done. 2. The mod 2 cohomology of BF4 The algebra structure of H*BF4 has been known since Borel: in [4], the coho- mology ring of H*F4 is calculated, and in view of [3, Th'eor`emes 17.3 and 19.2* *], the generators can be taken universally transgressive so it follows, Theorem 2.1. The mod 2 cohomology of BF4 is isomorphic to the polynomial algebra H*BF4 = F2[y4, y6, y7, y16, y24]. Moreover, Sq2y4 = y6, Sq1y6 = y7, and Sq8y16= y24. Nevertheless, the aim of this section is giving a more appropriate descripti* *on of H*BF4 as the ring of invariants on 5 variables under the action of a certain gr* *oup of block matrices. That description involves some understanding of the elementa* *ry abelian 2-subgroups of F4. Recall that: o Given a group G and a subgroup H, it is said that H is pure if all its elements, but the neutrum, are in the same G-conjugacy class. o Given an abelian group G, nG denotes the subgroup of elements of order n. Then the basic data about elementary abelian subgroups in F4 can be found in [1* *6, Tables I and VI, and Theorem 7.3]: Proposition 2.2. In F4 there are exactly two conjugacy classes of elements of order two listed below____________________________ |_Class_|_Centralizer___n|o._in_T | |__2A__|SU(2)_xZ=2Sp(3)_|_12____| |__2B__|___Spin(9)_____|___3____| Moreover: i)Any elementary abelian 2-subgroup of F4 is toral if and only if it does * *not contain a 2A-pure eights group. ii)There is, up to conjugacy, a unique maximal elementary abelian 2-subgroup represented, say, by E32 := <2T, `>, where ` is an involution in NF4T in- ducing -1 on T . Then, E32 has rank 5 and there is a 2B-pure subgroup ^E4of order 4 such that E32\E^4= E32\ 2A. Also, E32 = CF4(E32) and NF4(E32)=E32~=22.3: (GL 3(2) x 3). Indeed, the subgroup E32 F4 can be easily identified up to conjugation: Proposition 2.3. Consider the standard chain of inclusions G2 Spin(8) Spin(9) F4. Then the maximal elementary abelian 2-subgroup in F4 is (up to conjugation) V3 Z(Spin(8)) where V3 is the maximal (non toral) elementary abelian 2-subgroup in G2, and V3 Z(Spin(8)) \ TF4 = 2TF4. HOMOTOPY UNIQUENESS OF BF4 5 Proof.According to [34, Proposition 5.3], or [16, Table I and Theorem 6.1], the* *re exists V3 = (Z=2)3 maximal elementary abelian 2-subgroup of G2. As G2 is centerfree, then G2 \ Z(Spin(8)) = {1}, and Z(Spin(8)) CF4(G2). Hence V3 Z(Spin(8)) ~=(Z=2)5 and this group must be the maximal elementary abelian 2-subgroup of F4 (up to conjugation). Finally, the chain of inclusions G2 Spin(8) Spin(9) F4 induces the chain TG2 TSpin(8) TSpin(9) TF4, and as: o V3 \ TG2 = 2TG2 by [34, Propositions 5.3 and 5.4] and o Z(Spin(8)) TSpin(8) then (Z=2)4 ~=V3 Z(Spin(8)) \ TF4 = 2TF4. It is also possible to give the matrix representation of NF4(E32)=E32 Aut(E* *32) = GL 5(2): Proposition 2.4. For an appropriate choice of basis, the matrix representation * *of NF4(E32)=E32 Aut(E32) = GL 5(2) consists of matrices of the form (action on columns) ` ' GL 2(2) ** ** ** 0 GL 3(2) where each * = 0 or 1. Proof.The description of WF4(E32) := NF4(E32)=E32 GL 5(2) in Proposition 2.2 shows that WF4(E32) contains a 2-Sylow subgroup of GL 5(2). Therefore, up to conjugacy in GL 5(2), or equivalently, up to a change of base in E32, WF4(E3* *2) contains B, the subgroup of upper triangular matrices. Following [10, x65B and x65C], B is the standard Borel subgroup of GL 5(2), and as B WF4(E32), then WF4(E32) must be a standard parabolic subgroup of GL 5(2). There is 16 of such groups, and only two of them have the same order as WF4(E32) (hence one of them is WF4(E32)). Those two standard parabolic subgroups, namely P1 and P2, consist of matrices of the form (action on columns) ` ' _ * * ! GL 2(2) ** ** ** GL 3(2) ** ** 0 GL 3(2) or 0 GL2(2) respectively. But the conjugacy class distribution described in Proposition 2.2* *.ii), forces WF4(E32) = P1. The following theorem was communicated to the authors by C. Wilkerson: Theorem 2.5. Let E32= (Z=2)5 be a maximal elementary abelian 2-subgroup of F4, then H*BF4 = (H*BE32)WF4(E32). Therefore, the Steenrod algebra action on H*BF4 is given by the following table 6 ALE~S VAVPETI~C & ANTONIO VIRUEL _________________________________________________124816 |__x__|S|q_xS|q_x_|Sq_x_|_Sq_x____|____Sq__x_____|2 |__y4_|0|__|_y6__|_y4__|____0_____|______0_______| |__y6_|y|7_|_0___|y4y6_|____0_____|______0_______| |__y7_|0|__|_0___|y4y7_|____0_____|______0_______|22 |_y16_|0|__|_0___|_0___|y24+_y4y16_|____y16______|22 |_y24_|0|__|_0___|y4y24_|_y4y24___|y16y24+_y4y6y24_| Proof.By Proposition 2.2 and Lemma 3.1 there exists E32~=(Z=2)5 F4, maximal elementary abelian 2-subgroup, whose Weyl group in F4, WF4(E32), consists of matrices of type (action on columns) ` ' GL 2(2) ** ** ** 0 GL 3(2) where each * = 0 or 1. Therefore the action on H*BE32= F2[t1, t2, t3, t4, t5], * *where |ti| = 1, is given by the transposition of those matrices. Now we will calculat* *e the invariants H*(BE32)WF4(E32). Consider P := F2[z4, z6, z7, z16, z24] where subindex indicates degree and o z4, z6 and z7 are the Dickson invariants in the three variables t3, t4 a* *nd t5, o z16= a21+ a22+ a1a2, z24= a1a2(a1+ a2), being ai= t8i+ z4t4i+ z6t2i+ z7ti for i = 1, . .,.5. Notice that ai= 0 if i 6= 1, 2. It is clear that z4, z6, and z7 are invariants by the action of WF4(E32). No* *w we will show that z16and z24are invariants too. If M = [mi,j]1 i,j 52 WF4(E32) then (MT means "M transposed"): X5 X5 X5 X5 MT ai = ( mi,ntn)8 + z4( mi,ntn)4 + z6( mi,ntn)2 + z7( mi,ntn) = n=1 n=1 n=1 n=1 X5 = mi,n(t8n+ z4t4n+ z6t2n+ z7tn) = n=1 X5 = mi,nai= n=1 = mi,1a1 + mi,2a2, and therefore MT z16= (m1,1a1 + m1,2a2)2 + (m2,1a1 + m2,2a2)2+ + (m1,1a1 + m1,2a2)(m2,1a1 + m2,2a2) = = (m1,1+ m1,1m2,1+ m2,1)a21+ (m1,2+ m1,2m2,2+ m2,2)a22+ + (m1,2m2,1+ m1,1m2,2)a1a2 = = a21+ a22+ a1a2 = = z16. as the matrix [mi,j]1 i,j 22 GL2(F2) which implies m1,2m2,1+ m1,1m2,2= 1, m1,1+ m1,1m2,1+ m2,1= 1, m1,2+ m1,2m2,2+ m2,2= 1. HOMOTOPY UNIQUENESS OF BF4 7 For any matrix [mi,j]1 i,j 22 GL2(F2) it is also true that m1,1m2,1(m1,2+ m2,2) + m1,1+ m2,1= 1, m1,2m2,2(m1,1+ m2,1) + m1,2+ m2,2= 1, m1,1m2,1(m1,1+ m2,1)= 0, m1,2m2,2(m1,2+ m2,2)= 0, and therefore MT z24 = (m1,1a1 + m1,2a2)(m2,1a1 + m2,2a2)(m1,1a1 + m1,2a2 + m2,1a1 + m2,2a2) = = m1,1m2,1(m1,1+ m2,1)a31+ (m1,1m2,1(m1,2+ m2,2) + m1,1+ m2,1)a21a2+ + (m1,2m2,2(m1,1+ m2,1) + m1,2+ m2,2)a1a22+ m1,2m2,2(m1,2+ m2,2)a32= = a21a2 + a1a22= = z24. So we show that P H*(BE32)WF4(E32). As the ring F2[t1, . .,.t5] = H*(BE32) is an integral extension over P , by [37, Lemma 3.2], the degree is F2[t1, . .,.t5]:=Pdeg(z4) deg(z6) deg(z7) deg(z16) deg(z24) = = 210327. The degree of the extension W (E ) F2[t1, . .,.t5]: F2[t1, . .,.t5] F4 32 = |WF4(E32)| is 210327 [32, Theorem 79]. By the degree formula, F2[t1, . .,.t5]: P = W (E ) W (E ) = F2[t1, . .,.t5]: F2[t1, . .,.t5] F4 32 F2[t1, . .,.t5] F4 32* *:P and we see that W (E ) F2[t1, . .,.t5] F4 32:F2[y4, y6, y7, y16, y24] = 1. Thefore H*(BE32)WF4(E32)= F2[y4, y6, y7, y16, y24], and the Steenrod algebra structure can be read from [37, Corollary 2.4]. Finally, consider T a maximal torus in F4, then the diagram BiE32\T B E32\ T _______-BT | | | | BiT| | | |? BiE ?| BE32 _________32-BF4 is homotopy commutative, and o Bi*E32\Tis monomorphism, as by Proposition 2.3, E32\ T = 2T , o kerBi*Tis the ideal of H*BF4 generated by y7, hence kerBi*E32 kerBi*T= , which forces Bi*E32(y4) = z4, Bi*E32(y6) = Sq2Bi*E32(y4) = z6 and Bi*E32(y7) = Sq1Bi*E32(* *y6) = z7. Hence Bi*E32is injective and, as the Poincar'e series of H*BF4 and (H*BE32)WF4(* *E32) agree, H*BF4 ~=(H*BE32)WF4(E32). 8 ALE~S VAVPETI~C & ANTONIO VIRUEL An easy consequence is Corollary 2.6. Let E32= (Z=2)5 ____i-Spin(9) be the maximal elementary abelian 2-subgroup of Spin(9) described in Lemma 3.1. Then the cohomological morphism Bi* is injective. BiE32 Proof.The morphism BE32 ____Bi-BSpin(9) is obtained from BE32 _____- BF4 by taking centralizers of a rank 1 elementary abelian subgroup. Now calculating the cohomology of centralizers of elementary abelian subgroups is just applying Bi*E32 Lannes' T functor, which is exact. Therefore, as H*BF4 ____- H*BE32is injective Bi*=TOE(Bi*E32) by Theorem 2.5, then H*BSpin(9) = TOE(H*BF4) ___________- TOE(H*BE32) = H*BE32is injective, where TOEis the right component of T . 3. The Quillen category of F4 In this section we calculate the Quillen category of F4. First we recall the definition of Quillen category of a group. The Quillen category of a group G at a prime p, namely Qp(G), is defined as the category whose objects are pairs (V, ff) 2 Abx Mono(V, G) such that V is a non-trivial elementary abelian p-grou* *p, and with morphisms MorQp(G) (V1, ff1), (V2, ff2) , the set of group homomorphism f : V1 ! V2 such that (V1, ff1) = (V1, ff2f). The group of automorphisms in the Quillen category of an object (V, ff) is what, in Section cohomology-section, we called the Weyl group of ff(V ) in G. Another equivalent description of the Quillen category, that can be applied * *to the case of p-compact groups, can be found in [12]: given X a p-compact group, Qp(X) is the category whose objects consist of pairs (V, ff*) 2 AbxMono Ap(H*BX, H*BV* * ) such that V is a non-trivial elementary abelian p-group, and with morphisms Mor Qp(G)(V1, ff*1), (V2, ff*2) , the set of group homomorphism f : V1 ! V2 such that (V1, ff*1) = (V1, Bf*ff*2). Actually, we choose, for simplicity, a skeletal subcategory of Q2(F4); that * *is, a full subcategory containing just one representative for each isomorphism class * *of objects in Q2(F4). This election is described in Proposition 3.2. As the Weyl group index of Spin(9) in F4 is 3, any 2-subgroup of F4 lives (up to conjugation) in Spin(9). Therefore our calculations will be done in Spin(9). We use the description of the groups Spin(n) given in [9, Chapter 10]. Let {ei} denote a basis of the suitable Clifford algebra, then we fix some distingu* *ished elements in Spin(9). Define b1 = -1, b2 = e1e2e3e4e5e6e7e8, a1 = e1e2e3e4, a2 = e1e2e5e6, and a3 = e1e3e5e7. Let T Spin(9) be the preimage of the standard maximal torus in SO(9) with cover map Spin(9) ____æ-SO(9), that is 4 T= i=1e2i-1(cos(ti)e2i-1+ sin(ti)e2i) 2 Spin(9) 4 = i=1(- cos(ti) + sin(ti)e2i-1e2i) 2 Spin(9). This allow us to fix a representative of the maximal elementary abelian 2- subgroup of F4, as well as its Weyl group. From the proof of [16, Theorem 7.3] * *we get HOMOTOPY UNIQUENESS OF BF4 9 Lemma 3.1. The subgroup (Z=2)5 ~=V32= Spin(9) F4 is a representative of the maximal elementary abelian 2-subgroup of Spin(9) and F4, such that V32\T = . Moreover, this choice of base makes its* * Weyl group consists of matrices of the form (action on columns) ` ' GL 2(2) ** ** ** 0 GL 3(2) where each * = 0 or 1. Now, we can describe the Quillen category of F4. Proposition 3.2. The Quillen category of F4 at p = 2, Q2(F4), contains exactly 11 isomorphism classes of object with representatives listed below. All the rep* *resen- tatives are presented as subgroups of Spin(9) F4 ____________________________________________________________________ |___Class_Re|presentative_|_____Centralizer_________|Weyl____|_Toral_| |____2A____|_______|_____SU(2)_xZ=2Sp(3)______|__GL1(2)____|Yes_ | |____2B____|________|________Spin(9)__________|__GL1(2)____|Yes_3|1 |___4A____|______|_(S__xZ=2U(3))_o_Z=2_____|_GL2(2)____|Yes_2| |__4A_B__|_______|_Spin(4)_xZ=2Spin(5)_____|__Z=2_____|Yes__|3 |___4B____|______|_______Spin(8)__________|__GL2(2)____|Yes_7|2 |___8A____|____|__(Z=2)__x_O(3)________|_GL3(2)____|No__|6112 |__8A_B__|__(S_|xZ=2S__xZ=2U(2))_o_Z=2_|2_._GL2(2)_Y|es_4|32 |__8A_B___|____|Spin(4)_xZ=2Spin(4)____2|._GL2(2)__|Yes_1|433 |__16A__B__|__|(Z=2)__x_O(2)_______2|._GL3(2)__|No__|123 |_16A__B__|__V32\_T____|_________T_:__________W|F4=Z(WF4)_|Yes__|283 |_32A__B__|____V32_____|___________V32____________|_WF4(V32)___|No__| where the notation XAnBm means a group of order X such that n (resp. m) ele- ments are in the conjugacy class A (resp. B). Proof.Here we use the Dwyer-Wilkerson's approach to the Quillen category of G at the prime p, which means that Qp(G) can be read from the mod p cohomology of its classifying space. The group V32 is a maximal elementary abelian 2-subgroup and in Proposi- tion 2.5 we have prove that H*BF4 = (H*BV32)WF4(V32). Therefore, given any ff* 2 MonoA2(H*BF4, H*BV ) where V is an elementary abelian 2-subgroup, there exists f 2 Mono (V, V32) such that ff* = Bf*Bi*V32. This implies that the objec* *ts in Q2(F4) are just the orbits of the action of WF4(V32) on V32and the Weyl grou* *ps are the isotropy subgroups of WF4(V32). Centralizers are calculated by direct c* *om- putation in Spin(9) and in SU(2) xZ=2Sp(3). Remark 3.3. Notice that the representatives shown in Proposition 3.2 live in NSpin(9) N, and verify that the centralizer in NF4 is the normalizer of the ce* *n- tralizer in F4, that is, the representatives are shown as preferred lifts (see * *[21]) of their inclusions in F4. To finish this section, we define an interesting subcategory of Q2(F4). We c* *all QStub2(F4) the full subcategory of Q2(F4) whose objects are the representatives of the classes 2B, 4A2B, 4B3, 8A6B, 8A4B3, 16A12B3 and 32A28B3 listed in Proposition 3.2. Then 10 ALE~S VAVPETI~C & ANTONIO VIRUEL Remark 3.4. Notice that the objects of QStub2(F4) are all toral but V32, as it * *is shown in Proposition 3.2. Therefore, these toral objects are shown as the unique preferred lifts of their inclusions in F4, and any QStub2(F4)-automorphism of E, E 6= V32, can be realized as conjugation by an element in NF4 [25, Proposition * *4.1]. We also prove: Proposition 3.5. For any object (V, ff) in QStub2(F4), the self equivalences of* * the classifying space centralizer BCF4(ff(V )) are determined by restriction to its* * maxi- mal torus normalizer. Proof.Let (V, ff) be an object in QStub2(F4), and let CF4(ff(V )) and CF4(ff(V * *))0 be the centralizer of ff(V ) in F4 and its connected component respectively. As* * it is shown in the table in Proposition 3.2, CF4(ff(V ))0 is not of type SO(2n+1)xSp(* *n), hence by [19, Corollary 3.5], its self equivalences are determined by the restr* *iction to its maximal torus normalizer. Finally, as the center of CF4(ff(V ))0 equals the center of its maximal toru* *s nor- malizer, by [22, Proposition 4.2] the self equivalences of CF4(ff(V )) are also* * deter- mined by the restriction to its maximal torus normalizer. 4.The p-stubborn category versus Quillen category In this section we show that there exists an interesting connection between Rp(G), the p-stubborn category of G, and Qp(G). Finally we discuss this rela- tion in the particular case of F4 at p = 2. First we recall the definition of p-stubborn subgroup of a Lie group G. Let * *p be a fixed prime. A p-toral group is a compact Lie group P whose component of the unit, P0, is a torus and whose group of components P=P0, is a finite p-group. G* *iven a compact Lie group G, a p-toral subgroup P of G is said to be p-stubborn if the quotient NG (P )=P , where NG (P ) is the normalizer of P in G, is finite and d* *oes not contain any nontrivial normal p-subgroup. Therefore, the category Rp(G) is defi* *ned as the full subcategory of the orbit category whose objects are those orbits G=P for which P G is a p-stubborn subgroup. Notice that Rp(G) can also be thought as the category whose objects are P G, p-stubborn subgroups, and morphisms Mor Rp(G)(P1, P2) = P2=NG (P1, P2), where NG (P1, P2) = {g 2 G|gP1g-1 P2}. Therefore morphisms in Rp(G) can be thought as conjugations modulo action of the target group and we will denote them as cP2g. Given a Lie group G, the natural way of associating an elementary abelian p- subgroup of to a p-stubborn subgroup is via the center. In what follows, given a group P , pZ(P )denotes the elements of order p in the center of P (thus pZ(P )* *is an elementary abelian p-group). Theorem 4.1. Fix p a prime, and let G be a compact Lie group such that ß0G is a p-group. Then there exists a contravariant functor FG :Rp(G) _ Qp(G) given by FG (P ) = pZ(P )and FG (cP2g) = cg-1. Proof.Notice that for a p-stubborn subgroup P G, it holds that Z(P ) = CG (P ), since ß0G is a p-group [18, Lemma 1.5.(ii)]. Therefore cg(P1) P2 gives rise to Z(P2) = CG (P2) CG (cg(P1)) = Z(cg(P1)). Because cg is iso- morphism cg(Z(P1)) = Z(cg(P1)), and hence cg-1(Z(P2)) Z(P1). From this we get cg-1(pZ(P2)) pZ(P1). Finally, notice that if P2g0 = P2g, then g0 = qg for some q 2 P2, and therefore cg-1= cg0-1: pZ(P2)___- pZ(P1)as group morphisms. Thus FG is a contravariant functor between Rp(G) and Qp(G). HOMOTOPY UNIQUENESS OF BF4 11 Remark 4.2. The importance of Theorem 4.1 is that allow us to relate the two classical homology decompositions: via centralizers [17] and via p-stubborns [1* *8]. Indeed, given a Lie group G such that ß0G is a p-group, we can consider P ____iP-CG (pZ(P )) the natural inclusion and Theorem 4.1 shows that the diagram {BP }Rp(G)_____{BiP}-{CG (E)}Qp(G) is homotopy commutative. To finish this section, we consider the case of F4. Consider the skeletal su* *bcat- egory of R2(F4) given by those 2-stubborn subgroups in NF4 such that 2Z(P )is one of the representatives in Proposition 3.2, and denote it again by R2(F4), t* *hen Lemma 4.3. The image of R2(F4) by the functor FF4is a subcategory of QStub2(F4). Proof.As F4 is connected, for any P 2 R2(F4), it holds that Z(P ) = CF4(P ). Notice that Spin(9) F4, and by transfer arguments P Spin(9). Hence Z(Spin(9)) CSpin(9)(P ) CF4(P ) = Z(P ), and therefore Z(P ), and 2Z(P ), contains an element of class 2B. Finally, notice that there is no 2-stubborn subgroup P F4, whose center belongs to the class 16A14B, as in that case P (Z=2)3 x O(2), which implies P = (Z=2)3 x O(2) and NF4(P )=P = 23. GL3(2), which is not 2-reduced. Therefore, the 2-center of a 2-stubborn subgroup in F4 is either in the clas* *s 2B, 4A2B, 4B3, 8A6B, 8A4B3, 16A12B3 or 32A28B3. 5. The proof of Theorem 1.1 Throughout this section we prove Theorem 1.1. In what follows, - X is a 2-compact group whose maximal torus normalizer N ____j-X is isomorphic to that of F4, and - F4, Spin(9), CSpin(9)(E) and CF4(E) will denote the 2-compact group ob- tained from the 2-completion of the respective Lie groups. First we show that X is a connected 2-compact group as F4 is so. This is a consequence of the order of the normal subgroups of the Weyl group of X, WX , which is isomorphic to the Weyl group of F4. The order of these normal subgroups of WF4 are described in the following lemma: Lemma 5.1. The group WF4 has exactly twelve normal subgroups, and they have orders 1, 2, 32, 96, 96, 192, 192, 288, 576, 576, 576 and 1152. Proof.The proof was done by direct calculation by means of MAGMA [5], using the generators obtained from the Cartan matrix in [6, Planche VIII]. Proposition 5.2. X is connected. Proof.Assume that X is not connected and call X0 the connected component of the unit. Then there exists a short exact sequence: 0 ____- WX0 ____- WX ~=WF4 ____- ß0X ____- 0. As ß0X is a 2-group, WX0 must be a normal subgroup of WX ~=WF4 of index a power of 2. Lemma 5.1 shows that #|WX0| = 288, 576 or 1152. But according 12 ALE~S VAVPETI~C & ANTONIO VIRUEL to [8], no 2-adic pseudoreflection group of rank 4 has order 288 or 576, hence #|WX0| = 1152 and ß0X = 0. Thus X is connected. As X is connected, previous work by Dwyer-Wilkerson [15] allow us to determi* *ne the group component of centralizers in X so obtaining the isomorphism type of s* *ome centralizers Proposition 5.3. There exists a map f: BZ=2 ____- BN ____Bj-BX such that _ ev BNSpin(9)' map(BZ=2, BN)f___- BN | | | | | Bj*| |Bj | | | ?| OE ?| ev ?| BSpin(9) ' map(BZ=2, BX)f___- BX Proof.As N ~=NF4, we can identify Z=2 ~= N such that by Remark 3.3 CN (Z=2) = NSpin(9). Moreover, as X is connected, by [15, Theorem 7.6] we get t* *hat the Weyl group of CX (Z=2) agress with the Weyl group of its connected componen* *t, and then ß0CX (Z=2) = 0. Therefore CX (Z=2) is a connected 2-compact group with normalizer isomorphic to that of Spin(9). According to [29], CX (Z=2) ~=Spin(9) as 2-compact groups. Now, as the Weyl group index of Spin(9) in X is 3, by transfer arguments we know that the cohomology of BX injects into the cohomology of BSpin(9) via fSpin(9)= ev O OE, this allow us to determine H*BX. Proposition 5.4. There is an isomorphism of unstable algebras H*BX ~=A2H*BF4 induced by f*Spin(9), that is, Bi*Spin(9)(H*BF4) = f*Spin(9)(H*BX). Proof.Recall the situation: X is a 2-compact group whose maximal torus normal- izer N ____j-X is isomorphic to that of F4, we shall prove that H*BX ~=H*BF4 as unstable algebras over A2. According to Proposition 5.2, X is a connected 2-compact group, and it has t* *he same Weyl group as F4. Therefore H*Q^2BX = (H*Q^2BT )WX = (H*Q^2BT )WF4 = Q^2[q4, q12, q16, q24], and the Bockstein spectral sequence associated to H*BX converges to F2[q4, q12,* * q16, q24]. Now by Proposition 5.3, we know that there exists a commutative diagram: BNSpin(9)______-BN | | | | |Bj | | ?| fSpin(9)?| BSpin(9) ______-BX As the Weyl group index of Spin(9) in X is 3, by transfer arguments we know that the cohomology of BX injects into the cohomology of BSpin(9). Let V32 be as defined in Lemma 3.1. According to Corollary 2.6, H*BSpin(9) injects in H*BV32, thus H*BX does so. Therefore H*BX (H*BV32)WX(V32)where WX (V32) is the Weyl group of V32in X. Our next step is to determine WX (V32), or at least a big enough subgroup. T* *he Weyl group of V32 in Spin(9) can be read from Proposition 3.2, and consists of HOMOTOPY UNIQUENESS OF BF4 13 matrices of type (action on columns) `1 * * * *' 0 1 * * * 0 GL 3(2) where each * = 0 or 1. From Proposition 3.2, we can also calculate the Weyl gro* *up of V32in N, and it consists of matrices of type (action on columns) 0 1 GL2(2) ** ** 00 @ 0 GL 2(2) 00A 0 0 1 where again each * = 0 or 1. Therefore WX (V32) does contain the group generated by those two matrix groups, which is exactly WF4(V32). Hence H*BX (H*BV32)WX(V32) (H*BV32)WF4(V32)= H*BF4, that is H*BX is a subalgebra of H*BF4 and we showed its Bockstein spectral sequence converges to F2[q4, q12, q16, q24]. As the element q4 comes from something in * *di- mension 4, and H4BF4 = , then q4 = [y4]. Thus y4 2 H*BX, y6 = Sq2y4 2 H*BX, and y7 = Sq1y6 2 H*BX. As y26survives in the Bockstein spectral se- quence, then q12= [y26]. The element q16 must be a class coming from H16BX = H16BF4. As [y4y26] = [y4][y26] = q4q12 is independent of q16, t* *hen q16= [y16+ ~y4y26]. We already saw that y4y26is in H*BX, therefore y162 H*BX. Because Sq8y16= y24y16+ y242 H*BX and y24y162 H*BX, also the element y24 is in H*BX, which shows that H*BX = F2[y4, y6, y7, y16, y24] = H*BF4. Finally, notice that the isomorphism, namely ', is constructed such that the following diagram is commutative: f*Spin(9)* Bi*V32* H*BX ______-H BSpin(9)____-H BV32 | æ> | æ ' | æ * | æ BiSpin(9) ?|æ H*BF4 As Bi*V32is injective by Corollary 2.6, then Bi*Spin(9)(H*BF4) = f*Spin(9)(H*BX* *). A direct consequence of Proposition 5.4 and Lannes' theory is Corollary 5.5. Q2(X) is isomorphic to Q2(F4) as categories. The Quillen category of F4 at the prime 2 is calculated in Section 3 (actual* *ly a skeletal subcategory of Q2(F4) is calculated). We are interested in a full subc* *ategory of Q2(X), namely QStub2(X) ~=QStub2(F4), where the isomorphism is the restricti* *on of the isomorphism given in Corollary 5.5. By Proposition 5.4, the diagram {BE}QStub2(F4)__{fE}-BX fSpin(9) is commutative, where fE : BE ____- BNSpin(9)___- BSpin(9) ______- BX. Taking centralizers in X over the category QStub2(X) = QStub2(F4) we obtain the commutative diagram (5.1) {map (BE, BX)fE}QStub2(F4)__{ev}-BX. 14 ALE~S VAVPETI~C & ANTONIO VIRUEL Noticing that if E 2 QStub2(X) = QStub2(F4), then E = E . , by Lemma 7.10 in [15] we have that map (BE, BX)fE ' map(BE, map(B, BX)f)fE and using Proposition 5.3 we obtain (5.2) map(BE, BX)fE ' OE* ' map(BE, map(B, BX)f)fE' map (BE, BSpin(9))BiE ' ' BCSpin(9)(E) = BCF4(E) and (5.3) map(BE, BN)BiE ' _* ' map(BE, map(B, BN)Bi)BiE' map (BE, BNSpin(9))BiE ' ' BCNSpin(9)(E) = BNCF4(E) where the last equality comes from the fact that BE ____BiE-BN is a preferred l* *ift as showed in Remark 3.4. Now, the idea is to replace map (BE, BX)fE by BCF4(E) in diagram (5.1). Before doing so, we check the replacement will keep the diagram commutative (up to homotopy). As any QStub2(F4)-morphism E ____- E0 is composition of the fixed inclusion E-___- E0with a QStub2(F4)-automorphism E0 ___- E0, we have to worry only about QStub2(F4)-automorphism. If E = V32, then by Lannes' T functor map(BE, BX)fE ' BV32= BCF4(V32), and by Lannes' theory and Proposition 5.4 the diagram commutes up to homotopy. Now assume E 2 QStub2(X) = QStub2(F4) such that E 6= V32. Then by Remark 3.4 the inclusion E N is the only preferred lift of E in F4 (and therefore in* * X), and any QStub2(F4)-automorphism E ____h-E is induced by conjugation by g 2 N. So we have the homotopy commutative diagram BiE BE ______-BNw | ww | w h || www ?| BiE w BE ______-BN HOMOTOPY UNIQUENESS OF BF4 15 which together with diagrams (5.2)and (5.3)induces the homotopy commutative diagram OE* map (BE,|BX)fE _____________________________-BCF4(E)| | "Z æ> | | Z j æ | | Z * æ | | Z æ | | Z _* æ | || map(BE, BN)BiE __-BNCF4(E) || | | | | h* || h*|| _*(h*)|| |OE*(h*)| || ?| _ ?| || | map(BE, BN)BiE __*-BNCF4(E) | || æ Z || j æ Z | æ* Z | | æ Z | ?| =æ OE* Z~ ?| map (BE, BX)fE _____________________________-BCF4(E) Now, notice that h is just cg in N, hence we can replace _*(h*) by Bcg-1in t* *he diagram above and, as self equivalences of BCF4(E) are determined by restrictio* *n to BNCF4(E)by Proposition 3.5, we can also replace OE*(h*) by Bcg-1above. So gluing together this information with diagram (5.1)we obtain the homotopy commutative diagram {BCF4(E)}QStub2(F4)__{OE*}-{map (BE, BX)fE}QStub2(F4)__{ev}-BX, that composed with the commutative diagram constructed in Remark 4.2 gives rise to the (homotopy) commutative diagram (5.4) {BP }R2(F4)__________{evOOE*OBiP}-BX. Now, as one cannot take the hocolim of a diagram in the homotopy category, the diagram above does not ensure the existence of a map from hocolim{BP }R2(F4) (which is BF4 up to 2-completion) to BX. Such a map exists if some obstruction classes living in -limi+1----ßi(map (BP, BX)evOOE*OBiP) R2(F4) vanish [38, Proposition 3]. Here limiis the i-th derived functor of the inverse* * limit functor [7, Chapter XI,x6]. Proposition 5.6. For any i, and j, it holds limißj(map (BP, BX)fP) = 0, where fP = ev O OE* O BiP. Proof.Since every P contains = Z(Spin(9)), and Z(P ) then OE* map(BP, BX)fP ' map(BP, BCX ())OE*OBiP' OE* ' map(BP, BSpin(9))BiP ' BZ(P ), 16 ALE~S VAVPETI~C & ANTONIO VIRUEL and the induced map by Bcg behaves as Bcg-1by means of Theorem 4.1. Exactly the same description holds for map(BP, BF4)BiP. Hence -limi----ßj(map (BP, BX)fP)= -limi----ßj(map (BP, BF4)BiP) R2(F4) R2(F4) = 0 by [18]. So the obstruction classes vanish and the diagram (5.4)induces a map BF4 ___* *f-BX. To finish the proof of Theorem 1.1, we have to show that f induces an isomorphi* *sm of 2-compact groups. Notice that by construction the diagram BN2 ~=(BNSpin(9))2___-BN | | | | Bj| | | ?| f ?| BF4 ________-BX commutes, where N2 is the 2-normalizer of the maximal torus in F4 as well as the maximal 2-stubborn subgroup of F4. All the induced cohomological maps (but f*) are known to be injective, hence f* is so. As the Poincar'e series of H*BF4 and H*BX agree, f* is a cohomological isomorphism, thus a 2-compact group isomorphism. References [1]J. Aguad'e, Constructing modular classifying spaces, Israel J. Math. 66 (1* *989), 23-40. [2]K. Andersen, J. Grodal, J.M. Mfiller and A. Viruel, The classification of * *p-compact groups, p ood, in preparation. [3]A. Borel, Sur la cohomologie des espaces fibr'es principaux et des espace * *homog`enes de groupes de Lie compacts, Ann. Math. 57 (1953), 115-207. [4]A. Borel,Sur l'homologie et la cohomologie des groupes de Lie compacts con* *nexes, Amer. J. Math. 76 (1954), 273-342. [5]W. Bosma, J.J. Cannon, C. Playoust, The Magma Algebra System I: The user l* *anguage, J. Symbolic Computation 24 (1997), 235-265. [6]N. Bourbaki, Groupes et alg`ebres de Lie, 'El'ementes de math'ematique, He* *rmann, Paris (1968). [7]A. Bousfield and D. Kan, Homotopy limits, completion and localisation, SLN* *M 304, Springer Verlag (1972). [8]A. Clark and J. Ewing, The realization of polynomial algebras as cohomolog* *y rings, Pacific J. Math. 50 (1974), 425-434. [9]M.L. Curtis, Matrix groups, Springer-Verlag New York Inc. (1984). [10]M.L. Curtis and I. Reiner, Methods of Representation Theory, with applicat* *ions to finite groups and orders, II, Pure and Applied Math., John Wiley (1987). [11]W.G. Dwyer, Lie groups and p-compact groups, Proceedings of the Internatio* *nal Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, * *433-442. [12]W.G. Dwyer and C.W. Wilkerson, A cohomology decomposition theorem, Topolog* *y 31 (1992), 433-443. [13]W. Dwyer and C. Wilkerson, A new finite loop space at the prime two, J. Am* *er. Math. Soc. 6 (1993), 37-63. [14]W.G. Dwyer and C.W. Wilkerson, Homotopy fixed points methods for Lie group* *s and finite loop spaces, Ann. Math. 139 (1994), 395-442. [15]W.G. Dwyer and C.W. Wilkerson, The center of a p-compact group, The ~Cech * *Centennial. Contemporary Math. 181 (1995), 119-157. [16]R.L. Griess Jr., Elementary Abelian p-subgroups of Algebraic Groups, Geo. * *Dedicata 39 (1991) 253-305. HOMOTOPY UNIQUENESS OF BF4 17 [17]S. Jackowski and J. McClure, Homotopy decomposition of classifying spaces * *via elementary abelian subgroups, Topology 31 (1992), 113-132. [18]S. Jackowski, J. McClure and R. Oliver, Homotopy classification of self ma* *ps of BG via G actions, Ann. Math. 135 (1992) 183-270. [19]S. Jackowski, J. McClure, R. Oliver, Self homotopy equivalences of classif* *ying spaces of compact connected Lie groups, Fund. Math. 147 (1995), 99-126. [20]J.M. Mfiller, Homotopy Lie Groups, Bull. Amer. Math. Soc. 32 (1995), 413-4* *28. [21]J.M. Mfiller, Normalizers of maximal tori, Math. Z. 231 (1999), 51-74. [22]J.M. Mfiller, Deterministic p-compact groups, Fileds Institute Communicati* *ons 19 (1998), 255-278. [23]J.M. Mfiller, N-determined p-compact groups, Preprint June 21, 2000. [24]J. Mfiller and D. Notbohm, Centers and finite coverings of finite loop spa* *ces, J. Reine U. Angew. Math. 456 (1994), 99-133. [25]J.M. Mfiller and D. Notbohm, Connected finite loop spaces with maximal tor* *i, Trans. Amer. Math. Soc. 350 (1998), 3483-3504. [26]D. Notbohm, Fake Lie groups with maximal tori IV, Math. Ann. 294 (1992), 1* *09-116. [27]D. Notbohm, Classifying spaces of compact Lie groups and finite loop space* *s, Handbook of Algebraic Topology, North-Holland (1995). [28]D. Notbohm, Spaces with polynomial mod-p cohomology, Math. Proc. Cambridge* * Philos. Soc. 126 (1999), 277-292. [29]D. Notbohm, A uniqueness result for orthogonal groups as 2-compact groups,* * Preprint Jan- uary 6, 2000. [30]D. Notbohm, On the 2-compact group DI(4), Preprint January 10, 2000. [31]D. Quillen, On the cohomology and K-theory of the general linear group ove* *r a finite field, Ann. Math. 96 (1972), 552-586. [32]J. Rotman, Galois Theory, Second editiion, Springer-Verlag New York Inc. (* *1991). [33]A. Vavpeti~c and A. Viruel, Homotopy uniqueness of sympletic groups as 2-c* *ompact groups, in preparation. [34]A. Viruel, Homotopy uniqueness of BG2, Manuscripta Math. 95 (1998), 471-49* *7. [35]A. Viruel, Mod 3 Homotopy Uniqueness of BF4, to appear in J. Math. Kyoto U* *niv. [36]C. Wilkerson, Rational maximal tori, J. Pure Appl. Algebra 4, (1974), 261-* *272. [37]C. Wilkerson, A primer on the Dickson Invariants, Corrected version of the* * previous paper publish at Contemp. Math. 19 (1983), 421-434. [38]Z. Wojtkowiak, On maps from holim F to Z, in Algebraic Topology, Barcelona* * 1986, SLNM 1298, 227-236. [39]A. Zabrodsky, On the realization of invariant subgroups of i*(X), Trans. A* *mer. Math. Soc. 285 (1984), 467-496. Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija. Departamento de 'Algebra, Geometr'ia y Topolog'ia, Universidad de M'alaga, AP. 59, 29080 M'alaga, Spain.