HOMOTOPY FIXED POINT SETS AND ACTIONS ON HOMOGENEOUS SPACES OF p-COMPACT GROUPS by Kenshi Ishiguro and Hyang-Sook Lee Abstract. We generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups. Introduction. For a group ß, if X is a ß-space, the homotopy fixed point set Xhß is the set of ß-maps mapß (Eß, X). The fixed point set Xß embeds in Xhß as the subspace of constant maps. In [6], Dror Farjoun and Zabrodsky consider the relationship between Xß and Xhß , when ß is a finite group and X is a finite ß-simplicial complex. Our work has been motivated by a result of theirs. Namely, if ß is a finite p-group, they show that Xß is an empty set if and only if Xhß is empty. We observe that, when the finiteness condition on X is replaced by a p-local one, the corresponding result still holds. Recall [23, p557] that the mod p cohomological dimension of X, denoted by cdp(X), means the supremum of the integer m such that there exists a sheaf F of Z=pZ-modules with Hm (X; F ) 6= 0. If X is the p-completion of a finite complex, then cdp(X) < 1. Theorem 0. For a finite p-group ß, suppose a ß-space X is Fp-complete and cdp(X) is finite. Then Xß is an empty set if and only if the homotopy fixed point set Xhß is empty. Our proof is analogous to the one given by Dror Farjoun and Zabrodsky [6]. Their argument uses Miller's theorem (Sullivan Conjecture) [19], and we use its p-compact group version [9]. Our result is related to the generalized Sullivan Conjecture. Some results on this matter can be found in [4] and [5]. Typeset by AM S-TEX 1 2 A p-compact group, [8], is a loop space X such that X is Fp-finite and that its classifying space BX is Fp-complete. The p-completion of a compact Lie group G is a p-compact group if ß0(G) is a p-group. Suppose a map i : X ____-Y is a monomorphism of p-compact groups so that the homotopy fiber Y =X of the delooped map BX ____-BY is Fp-complete and Fp-finite. For a map f : Bß ____-BY , there is an extension map fe: Bß ____-BX if and only if (Y =X)hß 6= ;, [8, 3.4 Actions on homogeneous spaces]. BX ~fj3 |Bi j j |? Bß _______-fBY We consider what follows if (Y =X)hß 6= ;, and what conditions guarantee the existence of an extension map respectively. Theorem 1. Let X ____-Y be a monomorphism of p-compact groups and let ß be a finite p-group. Suppose the p-compact group X is abelian and f : Bß ____-BY is induced from a monomorphism. If there is an extension map Bß ____-BX, the group ß is abelian. Theorem 2. Suppose V is an elementary abelian p-group and T ____-Y is a maximal torus of a connected p-compact group Y with Weyl group W (Y ). If the map BT ____-BY induces H* (BY ; Fp) = H* (BT ; Fp)W (Y ), any map f : BV ____-BY has an extension map BV ____-BT . Let G be a compact Lie group and let H be a subgroup of G with inclusion i : H ____-G. In [6, Example D1], it is shown that, for a homomorphism æ : ß ____-G, the group æ(ß) is conjugate in G to a subgroup of H if and only if Bæ : Bß ____-BG lifts (up to homotopy) to Bß ____-BH. Of course the condition is equivalent to (G=H)æ(ß) 6= ;. We consider a generalization of this result. We recall that the G-action on G=H is based on the following two actions. A left action G x X ____-X and a right action X x K ____-X give us the G- space X=K. In the case of G=H, take X = G and K = H. In this particular case, the composite map G ____-G x X ____-X deloops to BG ____-BX, and similarly BK ____-BX, since the maps are induced by homomorphisms of groups. We consider the case when X is a sphere S2n-1 . The unitary group U (n) acts on the sphere S2n-1 from both left and right in the usual 3 way so that S2n-1 = U (n)=U (n - 1). Let a finite p-group ß is a subgroup of U (n) and K = S1 = U (1) ,! U (n). We note that S1 is the maximal torus of the mod p finite loop space S2n-1 when n divides p - 1. If the map ß ____-(S2n-1 )^p is a homomorphism of p-compact groups, Theorem 0 implies that the delooped map lifts to Bß ____-B(S1)^p if and only if (S2n-1 =S1)ß 6= ;. Theorem 3. If a finite p-group ß which is a subgroup of U (n) acting on S2n-1 as above is abelian, the fixed point set (S2n-1 =S1)ß is non-empty. If X ____-Y is a monomorphism of p-compact groups, and a finite p-group ß acts on both BX and BY , then (Y =X)hß is Fp-complete. A result of [8] implies that (Y =X)hß is Fp-finite. We consider the space (Y =X)hß which is obtained by the induced fibration over Bß with fiber Y =X from a map Bß ____-BY . Y =X=== = = = = =Y =X | | |? |? Eß xß (Y =X) _______-BX | | |? |? Bß ____________-BY Theorem 4. Let (Y =X)hß be the homotopy fixed point set obtained as above for a monomorphism X ____-Y of p-compact groups. If (Y =X)h~ is Fp-good for any subgroup ~ of the finite p-group ß, then (Y =X)hß is Fp-finite. Any nilpotent space is Fp-good, [3]. Assuming a nilpotency condition, the second author [18] shows the Fp-finiteness of a finite complex with an action of a p-compact toral group. The authors would like to thank Fukuoka University and Ewha Womans University for financial support. The first author would like to thank Em- manuel Dror Farjoun for his comments. The second author was supported by KOSEF 97-0701-02-01-5, partially supported by the MOST through R & D Program M10022040004-01G050900310. 4 0. A p-local generalization of the result of Dror Farjoun and Zabrodsky. The argument in this section is very similar to the one used in [6]. It can be considered as a p-local version. A compact topological group G is said to have the extended homotopy fixed point property (EHFPP) if for every Fp-complete G-space X with cdp(X) < 1, one has XG = ; if and only if XhG = ;. If G has no EHFPP, that is, there exist an Fp-complete G-space X with cdp(X) < 1 and XG = ;, and an equivariant map EG ____-X, we say that G is extended compressible. The following is an extended version of [6, Theorem B], and to show that (iii) implies (i), the result of [13, p45 Cor. 1] can be used as mentioned in [6]. Proposition 0.1. For an elementary abelian p-group V , suppose a V - space X is Fp-complete and cdp(X) is finite. Then the following are equiv- alent. (i)XV 6= ; (ii)XhV 6= ; (iii)The classifying map Ø : EV xV X ____-BV induces a monomorphism on mod p cohomology. The following is a p-local version of [6, Lemma 2.1] Lemma 0.2. Suppose the kernel H of an epimorphism G ____-G0 is a com- pact Lie group such that the loop space (BH)^p is a p-compact group. Let X be an Fp-complete and Fp-finite G0-complex. Then the natural compo- sition mapG0 (EG0, X) ____-mapG (EG, X) is a weak equivalence of spaces. Proof. The argument is very similar to the one used in [6]. Notice that mapG (EG, X) = mapG (EG=H, X) = mapG (BH, X), where H has a triv- ial action on X and EG is a free contractible H-space. We consider the diagram mapG (EG, X) _______-=mapG (EG=H, X) O6|E =| | _ |? mapG0 (EG0, X) _______-mapG (BH, X) To show OE is a weak equivalence, it suffices to show that _ is a weak equiv- alence. Taking the full function space from _, we get ~_: map(EG0, X) ____- map(BH, X). According to the result of [9, Theorem 9.3] we can see that 5 map(BH, X) ' X, since (BH)^p is a p-compact group and X is Fp- complete and Fp-finite. Now _~ is a G-map, and a weak homotopy equiv- alence when the G-actions are ignored. Therefore _~ induces a homotopy fixed point equivalence. Since EG0 and BH = EG=H are free G0-space, we see that mapG0 (BH, X) ' mapG0 (EG0, map(BH, X)) ' mapG0 (EG0, X) by [6, Lemma 2.2]. Consequently ~_must be a weak G0-equivalence. There- fore _ is weakly equivalent. It is well-known that, for a compact Lie group K, the loop space (BK)^p is p-compact if ß0(K) is a p-group. The problem on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group is considered in [15] and [16]. A result says that if (BK)^p is a p-compact group, then ß0K must be p-nilpotent. Let G be a group. Recall [12] that the Frattini subgroup of G is the intersection of maximal subgroups of G, denoted by (G). Let G0 = G= G be Frattini factor group of G. Using Lemma 0.2, the argument of the proof of [6, Theorem C] is applicable for the following result. Proposition 0.3. A finite p-group G is extended compressible if and only if its Frattini factor group G0 is extended compressible. Proof of Theorem 0. Since the Frattini factor of finite p-group is an ele- mentary abelian p-group, the desired result is immediate from Proposition 0.1 and Proposition 0.3. Let G be a p-compact toral group. It is known [8, Proposition 6.9] that any p-compact toral group G has a discrete approximation f : G1 ! G. For the discrete toral group G1 , there exists an increasing chain Gn Gn+1 . . .of finite subgroups of G1 such that G1 = [i n Gi. Then we have the following result. Corollary 0.4. Suppose X is an Fp-complete space with the proxy action of a p-compact toral group G and cdp(X) < 1. If G1 acts on X and XGi = ; for some finite p-subgroup Gi of G1 , then XhG is empty. Proof. Since XhG1 is equivalent to the homotopy inverse limit of the tower {XhGi } i n , if XGi = ; for some Gi then XhGi = ; by Theorem 0. This implies XhG1 = ;. According to [8, Proposition 6.8], the discrete approxi- mation f induces a homotopy equivalence XhG ! XhG1 . Therefore XhG is empty. Corollary 0.5. Let G1 be a p-discrete toral group. Suppose G1 -space X is Fp-complete and cdp(X) < 1. Then XhG1 = ; if and only if XG1 = ;. Proof. It suffices to show that XhG1 6= ; implies XG1 6= ;. So let XhG1 6= ;. Then there is m such that XhGi 6= ; for all i m. Theorem 0 says XGi 6= ; for all i m. Therefore XG1 6= ;. 6 Let Xhß denote the Borel construction so that Xhß = Eß xß X. Assume X satisfies the condition of Theorem 0. According to [8, Theorem 7.4] together with Theorem 0, we immediately conclude that Xhß is Fp-finite if and only if the fixed point set X~ is empty for any subgroup ~ of the finite p-group ß of order p. 1. Extension problems and the mod p cohomology. In this section we consider extension problems, and prove Theorem 1 and Theorem 2. Some results of mod p cohomology of classifying spaces will be used. Proof of Theorem 1. Recall that any abelian p-compact group is equiva- lent to the product of a p-compact torus and a finite abelian p-group, so that BX = (BG)^p for a compact abelian Lie group G, [9] and [21]. Thus the extension map f~ : Bß ____-BX is induced by a group homomorphism æ : ß ____-G, since ß is a finite p-group, [11]. It is enough to show that this group homomorphism æ is injective. Since f : Bß ____-BY is a monomor- phism, the induced homomorphism f * : H* (BY ; Fp) ____- H* (Bß; Fp) is finite, [8, Proposition 9.11]. This means that H* (Bß; Fp) is a finitely gen- erated module over f *(H* (BY ; Fp)). Consider the following commutative diagram H* (BX; Fp) f~i* i i ii) i 6|| H* (Bß; Fp) oe_____f*H* (BY ; Fp) Since f~*(H* (BX; Fp)) contains f *(H* (BY ; Fp)), we see that H* (Bß; Fp) is a finitely generated module over f~*(H* (BX; Fp)), and therefore f~* = ((Bæ)^p)* is finite. So a result of Quillen [23] implies that the kernel of æ is trivial. An argument analogous to the one used here shows the following result: Theorem 1.1. Let X ____-Y be a monomorphism of p-compact groups and let ß be a finite p-group. Suppose BX = (BG)^p for a compact Lie group G and f : Bß ____-BY is induced from a monomorphism. Assume there is an extension map Bß ____-(BG)^p so that this map is induced from a group homomorphism æ : ß ____-G. Then æ is injective. Proof of Theorem 2. Since H* (BY ; Fp) = H* (BT ; Fp)W (Y ), using a result of [2] one can show that there is a homomorphism OE : H* (BT ; Fp) ____- H* (BV ; Fp) which makes the following diagram commutative over the Steenrod algebra: 7 H* (BT ; Fp) OiEi i ii)i 6|| H* (BV ; Fp) oe_____f*H* (BY ; Fp) Here OE factors through the polynomial part of H* (BV ; Fp). We note that H* (B(Z=p)n ; Fp) = Fp[x1, . .,.xn ] (y1, . .,.yn ) for odd prime p where each yi has dimension 1 and each xi = fiyi has dimension 2. Since V is an elementary abelian p-group, a result of [17] implies that there is a homomorphism æ : V ____-T such that OE = (Bæ)*, and the following diagram is commutative: BT Bæ j3 | j j |? BV _______-fBY This completes the proof. For a connected compact Lie group G, we note that, for instance, if p is odd and G is p-torsion free, then H* (BG; Fp) is isomorphic to the invariant ring H* (BTG ; Fp)W (G) , where TG is a maximal torus, and W (G) denotes the Weyl group. Analogous results hold for connected p-compact groups X when H* (X; Z^p) is torsion free, [10] and [22]. Next we recall that SO(3) contains Z=2 Z=2 as a subgroup. Considering Theorem 2 when p = 2 and Y = SO(3)^2, we notice that H* (BSO(3); F2) 6~= H* (BS1; F2)Z=2, and that there is no extension for the monomorphism B(Z=2 Z=2) ____-BSO(3)^2, since rank(SO(3)) = 1. Generally, we see that if rank(G) < 2-rank(G), then H* (BG; F2) is not isomorphic to the invariant ring H* (BTG ; F2)W (G) . 2. Actions on homogeneous spaces and fixed point sets. We recall that a left action GxX ____-X and a right action X xK ____-X give the G-space X=K. Let [x] = xK 2 X=K. For g 2 G, the action is given by g . [x] = [gx]. If [x0] 2 (X=K)G , for any g we can find k 2 K such that gx0 = x0k. In the case G = U (n), X = S2n-1 and K = S1 = U (1) ,! U (n) as mentioned in the introduction, the equation of gx0 = x0k is expressed in the matrix form. For n = 2, for instance, the expression is given by the following: ` ' ` ' ~ ` ' ~T a11 a12 x1 z 0 a21 a22 . x2 = ( x1 x2) . 0 1 8 where AT denotes the transpose of a matrix A. Proof of Theorem 3. Since ß is abelian, all the irreducible representations of ß have degree 1. Consequently there is oe 2 U (n) such that oe-1 ßoe is a subgroup of T n, where T n is the maximal torus of U (n) consisting of diagonal matrices. Let g 2 ß and let g0 = oe-1 goe. If g0x = xk for some x 2 S2n-1 , then gx0 = x0k where x0 = oex. It remains to find such x 2 S2n-1 . Since g0 is a diagonal matrix, it is easy to see that if x = (1, 0, . .,.0) 2 S2n-1 , then, as seen below, 0 i1 0 1 0 1 1 2 0 i 1 3T 1 0 BB i2 CC BB0 CC 66 BB 1 CC77 @ ... A . @ ...A= 4 (1 0 . . . 0 ). @ ... A 5 0 in 0 0 1 for any g0 there is k 2 S1 such that g0x = xk. In Theorem 2, taking Y = (S2n-1 )^p when n divides p - 1, we obtain ((S2n-1 =S1)^p)hV 6= ;, and Theorem 0 says ((S2n-1 =S1)^p)V 6= ;. Note that, in general, if ß is a finite p-group and X is a finite ß-complex, then Xß 6= ; if and only if (X^p)ß 6= ;. This result follows from the following diagram Xß _________-(X^p)ß | | |? |? (Xß )^p _______-(X^p)hß and the result (Xß )^p' (X^p)hß , [20, Theorem 2]. Consequently it follows that (S2n-1 =S1)V 6= ;, which is a special case of Theorem 3 assuming the map V ____-(S2n-1 )^p is a homomorphism of p-compact groups. Next we consider the non-abelian case. Suppose Q8 denotes the quater- nion group in SU (2); Q8 =< a, b | a4 = 1, a2 = b2, bab-1 = a-1 > where ` ' ` ' a = i0 -0i , b = 01 -10 9 Taking x = (1, 0) as the base point of S3, the composite map Q8 ____- U (2) ____-U (2) x S3 ____-S3 is a homomorphism of groups. A direct calcu- lation shows (S3=S1)Q8 = ;. This result can be obtained from Theorem 1, since Q8 is non-abelian. We have the following generalization. Proposition 2.1. For n 2, let æ : ß ____-U (n) be an irreducible repre- sentation for a non-abelian finite p-group ß. Then (S2n-1 =S1)ß = ;. Proof. The center of a nontrivial finite p-group contains more than one element. Since the representation æ is irreducible, Schur's lemma tells us that we can find a 2 ß such that æ(a) is a diagonal matrix 0 i 0 1 BB i CC @ ... A 0 i where i is a p-th primitive root of unity. If (S2n-1 =S1)ß 6= ;, then an argument analogous to the one used in our proof of Theorem 3 shows that the following equation should be satisfied: 0 i 01 0 z1 1 2 0 i 0 1 3T BB i CC BBz2 CC 66 BB 1 CC77 @ ... A . @ ...A = 4 (z1 z2 . . . zn ). @ ... A 5 0 i zn 0 1 for suitable (z1, z2, . .,.zn ) 2 S2n-1 . Consequently it follows that zi = 0 for i = 2, . .,.n. Now let x0 = (z1, 0, . .,.0). Using the equations gx0 = x0k for all g 2 æ(ß), we see that all entries of the first column except the (1, 1)-entry of each matrix g are zero. This means that there would be a 1-dimensional invariant subspace. This is a contradiction, since the representation is irreducible and n 2. Therefore (S2n-1 =S1)ß = ;. We have seen that the G-action on G=H is based on the following two actions: GxX ____-X and X xK ____-X. In the case X = G, the composite of the based maps G ____-G x X ____-X deloops to BG ____-BX. Here we consider the deloopability problem for G = U (n) and X = S2n-1 or SU (n). Let _ : U (n) x S2n-1 ____-S2n-1 be the U (n)-action on S2n-1 . For this action, we will show that the p-completed map (U (n))^p ____-(S2n-1 )^p is not deloopable for any prime p. 10 _ Proposition 2.2. The composite map U (n) ____-U (n) x S2n-1 _____-S2n-1 is not deloopable at any prime p when n 2. Our proof for the case n 3 will use admissible maps, [1]. The case n = 2 is, however, treated separately. This is a special case of the following U (n)-action on SU (n). The action ~ : U (n) x SU (n) ____-SU (n) is given by 0 1 1 0 B ... C ~(A, B) = A . B . B@ CA 1 0 det A-1 for A 2 U (n) and B 2 SU (n). This action is transitive, and the isotropy subgroup at the identity is isomorphic to U (1). Proposition 2.3. The composite map U (n) ____-U (n) x SU (n) _____-~SU (n) is not deloopable at any prime p. Proof. There is a finite covering Z=n ____- SU (n) x S1 _____-qU (n), where q|SU(n) is the inclusion SU (n) ,! U (n) and q(S1) is the center of U (n). If the map U (n) ____-SU (n) is deloopable at p, we obtain a map induced from the composition (BSU (n))^px (BS1)^p____-(BSU (n))^p The axis (BSU (n))^p____-(BSU (n))^pis the identity map. According to [14, Theorem 1], the other axis (BS1)^p ____-(BSU (n))^p should factor through (BZ=n)^p, where Z=n is the center of SU (n). This means that the map (BS1)^p ____-(BSU (n))^p would be a zero map. This is a contradiction, since the map S1 ____-SU (n) is a monomorphism. Thus we obtain the desired result. Lemma 2.4. The two U (2)-spaces SU (2) and S3 are U (2)-homeomorphic. Proof.iAjhomeomorphism ø : SU (2) ____- S3 is given by a map sending a -~b to a where a, b 2 C with |a|2 + |b|2 = 1. The desired result b ~a b follows from the following commutative diagram: U (2) x SU (2) _______-~SU (2) 1xø | ø | |? _ |? U (2) x S3 ___________-S3 This completes the proof. Proof of Proposition 2.2. The case n = 2 is proved by Proposition 2.3 and Lemma 2.4. So we assume n 3. If the map U (n) ____-S2n-1 was 11 deloopable at p, we would have a map BU (n)^p ____-B(S2n-1 )^p. Notice that the Lie group SU (n) is simple, and rank(SU (n)) 2. According to [1, Proposition 2.12], the restriction of the delooped map on BSU (n)^p is null homotopic. BU (n)^p _______-B(S2n-1 )^p 6| * | BSU (n)^p On the other hand, the restriction of U (n) ____-S2n-1 on U (1) is not null homotopic. BU (n)^p _______-B(S2n-1 )^p 6| 6| | | BU (1)^p _________-idB(S1)^p Consequently the map BSU (n)^p____-B(S2n-1 )^p should be essential, since U (1) ,! SU (2) ,! SU (n). This contradiction completes the proof. 3. Some properties of homotopy fixed point sets (Y =X)hß . Suppose that X ____-Y is a homomorphism of p-compact groups, that a finite p-group ß acts on classifying spaces BX and BY , and that BX ____- BY is a ß-map. According to [8, Lemma 10.6 and Proposition 5.8], if BY hß 6= ;, then (Y =X)hß 6= ; and the space is Fp-complete. If the map X ____-Y is a monomorphism, then Y =X is Fp-finite. We see, by [8, Theo- rem 4.6], that (Y =X)hß is Fp-finite. As mentioned in the introduction, next we consider the space (Y =X)hß which is obtained by the induced fibration over Bß with fiber Y =X from a map Bß ____-BY , [8, Lemma 10.4]. Y =X=== = = = = =Y =X | | |? |? Eß xß (Y =X) _______-BX | | |? |? Bß ____________-BY A space X is said to be Fp-good, [3], if H*(X; Fp) ____-H*(X^p; Fp) induced from the Fp-completion map X ____-X^p is an isomorphism. For instance, it is known [3, Ch VII Proposition 5.1] that if the fundamental group ß1X is finite, then X is Fp-good for any prime p. 12 Proof of Theorem 4. Recall [8, Remark 11.13] that a space is Fp-complete if and only if X is both Fp-local and Fp-good. Since Y =X is Fp-local, so is (Y =X)h~ for any subgroup ~ of the finite p-group ß. From our assumption, we see that each (Y =X)h~ is Fp-complete. 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