Homotopy type and v1-periodic homotopy groups of p-compact groups
aa r X i v : . [ m a t h . A T ] O c t HOMOTOPY TYPE AND v -PERIODIC HOMOTOPY GROUPS OF p -COMPACT GROUPS DONALD M. DAVIS
Abstract.
We determine the v -periodic homotopy groups of allirreducible p -compact groups ( BX, X ). In the most difficult, mod-ular, cases, we follow a direct path from their associated invariantpolynomials to these homotopy groups. We show that, if p is odd,every irreducible p -compact group has X of the homotopy type ofa product of explicit spaces related to p -completed Lie groups. Introduction
In [4] and [3], the classification of irreducible p -compact groups was completed.This family of spaces extends the family of ( p -completions of) compact simple Liegroups. The v -periodic homotopy groups of any space X , denoted v − π ∗ ( X ) ( p ) , area localization of the portion of the homotopy groups detected by K -theory; they weredefined in [20]. In [17] and [16], the author completed the determination of the v -periodic homotopy groups of all compact simple Lie groups. Here we do the same forall the remaining irreducible p -compact groups. Recall that a p -compact group ([22]) is a pair ( BX, X ) such that BX is p -completeand X = Ω BX with H ∗ ( X ; F p ) finite. Thus BX determines X and contains morestructure than does X . The homotopy type and homotopy groups of X do not takeinto account this extra structure nor the multiplication on X . We show that, if p is Date : October 25, 2007.2000
Mathematics Subject Classification.
Key words and phrases. v -periodic homotopy groups, p -compact groups, Adamsoperations, K -theory, invariant theory.The author is grateful to Kasper Andersen, John Harper, Hirokazu Nishinobu,and Clarence Wilkerson for valuable suggestions related to this project, each ofwhich is specifically described in the paper. If the groups v − π i ( X ) are finite, then p -completion induces an isomorphism v − π ∗ ( X ) → v − π ∗ ( X p ). ([9, p.1252]) odd, every irreducible p -compact group has X of the homotopy type of a product ofexplicit spaces related to p -completed Lie groups.According to [4, 1.1,11.1] and [3], the irreducible p -compact groups correspond tocompact simple Lie groups and the p -adic reflection groups listed in [2, Table 1] forwhich the character field is strictly larger than Q . See [13, pp.430-431] and [27, p.165]for other listings of reflection groups. We use the usual notation (( BX n ) p , ( X n ) p ),where n is the Shephard-Todd numbering ([36] or any of the previously-mentionedtables) and p is the prime associated to the completion.We will divide our discussion into four families of cases:(1) The compact simple Lie groups—infinite family 1, part of in-finite family 2, and cases 28, 35, 36, 37 in the Shephard-Toddlist.(2) The rest of the infinite families numbered 2a, 2b, and 3.(3) The nonmodular special cases, in which p does not divide theorder of the reflection group. This is cases 4-27 and 29-34.(4) The modular cases, in which p divides the order of the reflec-tion group. These are cases ( X ) , ( X ) , ( X ) , ( X ) , and( X ) . (Actually, we include ( X ) in Case (3) along with thenonmodular cases, and the Dwyer-Wilkerson space ( X ) washandled in [6].)Here is a brief summary of what we accomplish in each case. The author feels thathis contributions here are nil in case (1) , minuscule in case (2), modest in case (3),and significant in case (4).(1) Spaces X , X , X , X , and X are, respectively, SU ( n ), F , E , E , and E . These are p -compact groups for all primes p , although for small primes they were excluded by Clark andEwing ([13]) because H ∗ ( BX ; F p ) is not a polynomial algebra.The exceptional Lie group G is the case m = 6 in infinite family2b. The spaces SO ( n ), Spin( n ), and Sp ( n ) appear in the infinite Cases in which distinct compact Lie groups give rise to equivalent p -compactgroups are discussed in [4, 11.4]. But he accomplished much in these cases in earlier papers such as [16], [17],and [18]. -COMPACT GROUPS 3 family 2a with m = 2. Simplification of the homotopy typesof many of these, when p is odd, to products of spheres andspherically-resolved spaces was obtained in [31, (8.1),8.1]. The v -periodic homotopy groups of these spaces were computed in[18], [7], [17], [16], [8], and other papers. We will say no moreabout these cases.(2) In Section 2, we use work of Castellana and Broto-Moller toshow that the spaces in the infinite families can be decomposed,up to homotopy, as products of factors of p -completions of uni-tary groups, spheres, and sphere bundles over spheres. See 2.3,2.5, and 2.7 for the specific results.(3) In Table 3.2, we list the homotopy types of all cases ( X n ) p whichare not products of spheres. There are 31 such cases. In eachcase, we give the homotopy type as a product of spheres andspaces which are spherically resolved with α attaching maps.In Remark 3.3, we discuss the easily-computed v -periodic ho-motopy groups of these spaces.(4) The most novel part of the paper is the determination of the v -periodic homotopy groups of ( X ) , ( X ) , and ( X ) . Weintroduce a direct, but nontrivial, path from the invariant poly-nomials to the v -periodic homotopy groups. En route, we de-termine the Adams operations in K ∗ ( BX ; ˆ Z p ) and K ∗ ( X ; ˆ Z p ).In the case of ( X ) , we give new explicit formulas for the in-variant polynomials. We prove in 4.15 (resp. 5.17) that thehomotopy type of ( X ) (resp. ( X ) ) is directly related to SU (20) (resp. SU (42)). We prove in 4.20 that ( X ) has thehomotopy type of the 5-completion of a factor of E . Theselatter homotopy-type results and their proofs were pointed outand explained to the author by John Harper. DONALD M. DAVIS Infinite families 2 and 3
Family 3 consists of p -completed spheres S m − with p ≡ m , which is aloop space due to work of Sullivan ([37]). The groups v − π ∗ ( S m − ) ( p ) , originally dueto Mahowald ( p = 2) and Thompson ( p odd), are given in [19, 4.2].Family 2 consists of spaces X ( m, r, n ) where m > r | m , and n >
1. The “de-grees” of X ( m, r, n ) are m, m, . . . , ( n − m, mr n . These are the degrees of invariantpolynomials under a group action used in defining the space. The Clark-Ewing tabledoubles the degrees to form the “type,” as these doubled degrees are the degrees ofgenerators of H ∗ ( BX ; F p ) in the cases which they consider. For most of the irre-ducible p -compact groups X , H ∗ ( X ; F p ) is an exterior algebra on classes of grading2 d −
1, where d ranges over the degrees. Family 2b consists of spaces X ( m, r, n ) inwhich n = 2 and r = m , while family 2a is all other cases. The reason that these areseparated is that 2b has more applicable primes. Indeed, for family 2a, there are p -compact groups when p ≡ m , while for family 2b, these exist when p ≡ ± m , and also p = 2 if m = 4 or 6, and p = 3 if m = 3 or 6. The case m = 6 in family2b is the exceptional Lie group G . Note that all primes work when m = 6. The case( p = 2 , m = 4) has X = Sp (2), while ( p = 3 , m = 3) has X = SU (3) or P SU (3), theprojective unitary group. In this case, there are two inequivalent p -compact groupscorresponding to the same Q p -reflection group; however, since SU (3) → P SU (3) is a3-fold covering space, they have isomorphic v -periodic homotopy groups.The following results of Broto and Moller ([11]) and Castellana ([12]) will be useful.They deal with the homotopy fixed-point space X hG when G acts on a space of thesame homotopy type as a space X . Here and throughout, C m denotes a cyclic groupof order m , and U ( N ) is the p -completion of a unitary group. Theorem 2.1. ([11 , . , . If m | ( p − , ≤ s < m , and n > , then U ( mn + s ) hC m ≃ X ( m, , n ) and is a factor in a product decomposition of U ( mn + s ) . All of our spaces are completed at an appropriate prime p . This will not alwaysbe present in our notation. For example, we will often write SU ( n ) when we reallymean its p -completion. According to [34], the only exclusions are certain compact Lie groups when p is very small. -COMPACT GROUPS 5 Theorem 2.2. ([11 , . , . If m | ( p − , m ≥ , r > , and n ≥ , then X ( m, r, n ) hC m ≃ X ( m, , n − and is a factor in a product decomposition of X ( m, r, n ) . Corollary 2.3. If m | ( p − and r > , then X ( m, r, n ) ≃ X ( m, , n − × S n mr − and X ( m, , n − is a factor in a product decomposition of U ( m ( n − . Here X ( m, ,
1) is interpreted as S m − . Proof.
We use Theorem 2.2 to get the first factor. By the Kunneth Theorem, theother factor must have the same F p -cohomology as S n mr − , and hence must have thesame homotopy type as this sphere. Now we apply Theorem 2.1 to complete theproof. Remark 2.4.
Our Corollary 2.3 appears as [12, 1.4], except that she has an apparenttypo regarding the dimension of the sphere. Also, neither she nor [11] have therestriction r >
1, but it seems that the result is false for r = 1, since by induction itwould imply that X ( m, , n ) is a product of spheres, which is not usually true. Remark 2.5.
Let p be odd. By [31], for any N , p -completed SU ( N ) splits as aproduct of ( p −
1) spaces, each of which has H ∗ ( − ; F p ) an exterior algebra on odddimensional classes of dimensions b , b + q, . . . , b + tq , for some integers b and t . Hereand throughout q = 2( p − X ( m, , n −
1) will be a product of ( p − /m of these spaces for SU ( m ( n − v -periodic homotopy groups of these spacescan be read off from those of SU ( m ( n − p −
1) factors have v -periodichomotopy groups in nonoverlapping dimensions. Thus, to the extent that [18] isviewed as being a satisfactory description of v − π ∗ ( SU ( n )) ( p ) , Corollary 2.3 gives v − π ∗ ( X ( m, r, n )) ( p ) provided m | ( p − [18, 1.4] states that v − π k ( SU ( n )) ( p ) is a cyclic p -group with exponentmin( ν p ( j ! S ( k, j )) : j ≥ n ), where S ( − , − ) denotes the Stirling number of the secondkind. In [21], more tractable formulas were obtained if n ≤ p − p + 1. Here andthroughout, ν p ( − ) is the exponent of p . DONALD M. DAVIS
Example 2.6.
Let p = 7 . Then X (2 , , ≃ X (2 , , × S . There is a productdecomposition ( SU (10)) ≃ B (3 , × B (5 , × B (7 , × S × S × S , where B (2 n + 1 , n + 13) denotes a 7-completed S n +1 -bundle over S n +13 with at-taching map α . Then X (2 , , ≃ B (3 , × B (7 , × S . What remains for Family 2 is the cases 2b when m | ( p + 1). These are the spaces X ( m, m,
2) with m | ( p + 1). Let B (3 , p + 1) denote the p -completion of an S -bundleover S p +1 with attaching map α . Theorem 2.7. If m | ( p + 1) , then X ( m, m, ≃ B (3 , p + 1) m = p + 1 S × S m − m < p + 1 . Proof.
Let X = X ( m, m,
2) with m | ( p + 1). Then H ∗ ( X ; F p ) = Λ[ x , x m − ]. If m < p + 1, then by the unstable Adams spectral sequence ([10]), both classes x and x m − are spherical. Indeed, the E -term begins with towers in dimensions 3and 2 m − X is an H -space, the maps S → X and S m − → X yield a map S × S m − → X , and it is a p -equivalence by Whitehead’s Theorem.On the other hand, suppose m = p + 1. We will show that P ( x ) = x p +1 . It thenfollows from [31, 7.1] that there is a p -equivalence B (3 , p + 1) → X .To see that P ( x ) = x p +1 , we use the classifying space BX , which satisfies H ∗ ( BX ; F p ) = F p [ y , y p +2 ]. We will prove that P ( y ) = y p +2 + Ay ( p +1) / , forsome generator y p +2 and some A ∈ F p , from which the desired result about the x ’sfollows immediately from the map Σ X → BX , which in H ∗ ( − ; F p ) sends y j +1 to x j and sends products to 0.First note that P ( y ) = Ay ( p +1) / + By p +2 P ( y p +2 ) = Cy p + Dy ( p − / y p +2 , -COMPACT GROUPS 7 for some A , B , C , D in F p . By the unstable property of the Steenrod algebra, P p +1 ( y p +2 ) = y p p +2 . (2.8)We must have P p ( y p +2 ) = p − X j =0 c j y j ( p +1) / y p − − j p +2 , for some c j ∈ F p . Since P p +1 = P P p and P ( y i y j p +2 ) = i P ( y ) y i − y j p +2 + jy i P ( y p +2 ) y j − p +2 , (2.9)the only way to obtain (2.8) is if c B = 1 in F p . Thus B must be a unit, and thegenerator y p +2 can be chosen so that B = 1.3. Nonmodular individual cases
In this section, we consider all cases 4 through 34, excluding case 28 (which is F ),in the Shephard-Todd numbering in which p does not divide the order of the reflectiongroup. We obtain a very attractive result. One modular case, ( X , p = 3) is alsoincluded here. There is some overlap of our methods and results here with those in[29]. Theorem 3.1.
Let X = ( X n ) p with ≤ n ≤ and n = 28 , excluding the modularcases ( X ) , ( X ) , and ( X ) , which will be considered in the next two sections.Then X ≃ Q S d − , where d ranges over the integers listed as the “type” in [13] ,except for the 31 cases listed in Table 3.2. In these, each B ( − , . . . , − ) is built byfibrations from spheres of the indicated dimensions, with α as each attaching map,and occurs as a factor in a product decomposition of the p -completion of some SU ( N ) . We will call the integers d , which are 1/2 times the “type ” numbers of Clark-Ewing,the “degrees.” DONALD M. DAVIS
Table 3.2. Cases in Theorem 3.1 which are not products of spheres
Case Prime Space B (11 , B (15 , B (15 , B (23 , B (11 , B (11 , B (39 , B (39 , B (59 , B (23 , B (7 , × S
25 7 B (11 , × S
26 7 B (11 , , B (11 , × S
27 19 B (23 , × S
29 13 B (15 , × S × S
29 17 B (7 , × S × S
30 11 B (3 , × B (39 , B (3 , × S × S
30 29 B (3 , × S × S
31 13 B (15 , × B (23 , B (15 , × S × S
32 7 B (23 , , , B (23 , × B (35 , B (23 , × S × S
33 7 B (7 , × B (11 , , B (11 , × S × S × S
34 13 B (11 , , , × B (23 , B (11 , , × B (23 , × S
34 31 B (23 , × S × S × S × S
34 37 B (11 , × S × S × S × S Remark 3.3.
The v -periodic homotopy groups of B (2 n + 1 , n + 2 p −
1) wereobtained in [8, 1.3]. Those of B (11 , , and B (23 , , , were obtained in[8, 1.4]. Using [21, 1.5,1.9], we find that for ǫ = 0 , v − π t − ǫ ( B (11 , , , (13) ≈ t Z / max( f ( t ) ,f ( t ) ,f ( t ) ,f ( t )) t ≡ , -COMPACT GROUPS 9 where f γ ( t ) = min( γ, ν ( t − γ )), while v − π t − ǫ ( B (11 , , (19) ≈ t Z / max( f ′ ( t ) ,f ′ ( t ) ,f ′ ( t )) t ≡ , where f ′ γ ( t ) = min( γ, ν ( t − γ )). Proof of Theorem 3.1.
It is straightforward to check that the pairs (case, prime) listedin Table 3.2 are the only non-modular cases in [2, Table 1] in which an admissibleprime p satisfies that ( p −
1) divides the difference of distinct degrees. Indeed allother admissible primes have ( p −
1) greater than the maximum difference of degrees.For example, Case 30 requires p ≡ , p −
1) can. Thus the unstableAdams spectral sequence argument used in proving Theorem 2.7 works the same wayhere to show that X is a product of S d − in all cases not appearing in Table 3.2.In the relevant range, the E -term will consist only of infinite towers, one for eachgenerator. The first deviation from that is a Z /p in filtration 1 in homotopy dimension(2 d −
1) + (2 p − d is the smallest degree. This will always be greater thanthe dimension of the largest S d − .The next step is to show that the Steenrod operation P in H ∗ ( X ; F p ) must connectall the classes listed as adjacent generators in one of the B -spaces in Table 3.2. Thiswas achieved independently of, and slightly earlier than, the author in [33] and [25].We include several of our proofs, omitting the most complicated cases, to illustrateour methods and for the benefit of the reader without access to [33] and [25]. Weaccomplish this by considering the A -module H ∗ ( BX ; F p ). With one exception , allcases involving factors of B (2 m − , m +2 p −
3) are implied by Lemma 3.9 by applying H ∗ ( BX ) → H ∗− ( X ), which sends products to 0. Similarly, Lemma 3.11 covers thetwo cases with a factor B (11 , , X have the homotopy type claimed. Thefirst 10 cases are immediate from [31, 7.1], and the two other non-product cases,i.e. ( X ) and ( X ) , follow from [31, 7.2,7.6]. Note that these results of [31] did The exception is ( X ) , which is handled in [33]. We thank Nishinobu forpointing out a gap in the argument for this case which appeared in an earlierversion of this paper. not deal with p -completed spaces, but the obstruction theory arguments used thereapply in the p -complete context. There are 15 additional types which we claim to bequasi p -regular. As defined in [32], a space is quasi p -regular if it is p -equivalent to aproduct of spheres and spaces of the form B (2 n + 1 , n + 2 p − p -regular (forappropriate p ) using a skeletal approach. We could use that approach here, but weprefer to use the unstable Adams spectral sequence (UASS). The two methods arereally equivalent.Let q = 2 p −
2. In Diagrams 3.4 and 3.5, we illustrate the UASS for S n +1 indimension less than 2 n + pq − B (2 n + 1 , n + q + 1) in dimension less than2 n + 3 q −
3. Diagram 3.4 gives a nice interpretation of the statement of the homotopygroups in [38, 13.4]. If n ≥ p , the paired dots in Diagram 3.4 will not occur in thepictured range. The nice thing about these charts is that the F p -cohomology groupsof our spaces X are known to agree with that of their putative product decompositionas unstable algebras over the Steenrod algebra, and are of the required universal formfor the UASS to apply; hence their UASS has E -term the sum of the relevant chartsof spheres and B -spaces. In all cases, there will be no possible differentials.One can check that in all 15 cases in which X is claimed to be quasi p -regular,the towers in UASS( X ) corresponding to the spheres and the bottom cell of each B (2 n + 1 , n + q + 1) cannot support a differential, and hence yield maps from thesphere or S n +1 into X . Next one checks that π n + q ( X ) = 0 and π n + q +1 ( X ) = 0.As these are the groups in which the obstruction to extending the map S n +1 → X over B (2 n + 1 , n + q + 1) lie, we obtain the desired extension. Finally, we take theproduct of maps B → X and S d i − → X , using the group structure of X , to obtainthe desired p -equivalence from a product of spheres and B -spaces into X .The remaining cases, ( X ) , ( X ) , and ( X ) , are handled similarly. The E -term of the UASS converging to π ∗ ( X ) is isomorphic to that of its putative productdecomposition. For example, E ( X ) is the sum of Diagram 3.5 with n = 11and q = 24 plus Diagram 3.6. We can map S → X and S → X corresponding togenerators of homotopy groups. Then we can extend the first map over the 47- and 70-cells because π ( X ) = 0 and π ( X ) = 0. This gives a map B (23 , → X . Similarlywe can extend the second map over cells of B (11 , , ,
83) of dimension 46, 70, 94, -COMPACT GROUPS 11 X , yields the desired 13-equivalence B (23 , × B (11 , , , → X . The other two cases are handled similarly. Diagram 3.4.
UASS ( S n +1 ) in dim < n + pq − . Here n < p . ✻ rrrr r r ♣ ♣ ♣ r r r r r♣ ♣ ♣ n +1 2 n + q n +2 q n + nq n +( n +1) q n +( p − q Where there is a pair of dots, the grading at the bottom refers to the one on right,and the other is in grading 1 less.
Diagram 3.5.
UASS ( B (2 n + 1 , n + q + 1)) in dim < n + 3 q − . ✻ ✻ rrr rrr rrr (cid:0)(cid:0)(cid:0)✠ rrr (cid:0)(cid:0)✒ if n ≤ if n = 1 n +1 2 n + q +1 2 n +2 q n +3 q Diagram 3.6.
UASS ( B (11 , , , ) in dim < . ✻ ✻ ✻ ✻ rrr rrr rrr rrr rrrr rrrr rrrr rrrrr r
11 35 59 83 106 130 154 178
An alternate proof of Theorem 3.1 can be obtained using [15, 1.3,1.4], which canbe interpreted as the following theorem. In Section 6, we will provide a proof of astrengthened version of this result.
Theorem 3.7. ([15 , . , . If X is an H -space of rank r < p − with torsion-freehomology, then there are H -spaces X , . . . , X r with X = S n and X r = X , and thereare fibrations X i − → X i → S n i for ≤ i ≤ r as in the diagram X −−−→ X −−−→ · · · −−−→ X r y y S n S n r . (3.8) The homotopy type of the p -localization of X is determined by the elements of π n i − ( X i − ) associated to these fibrations. In order to apply this, one would still need to determine the P -action and to checkthat the relevant homotopy groups π n i − ( X i − ) are cyclic.In the following lemmas, which were used above, g i denotes a generator in grading i . Lemma 3.9. a. If m mod p and F p [ g m , g m +2 p − ] is an unstable A -algebra, then P g m ≡ ug m +2 p − mod decomposables, with u = 0 . -COMPACT GROUPS 13 b. The same conclusion holds if the unstable A -algebra contains additional gener-ators in dimensions d m mod (2 p − , provided also that d + 2 p − mod (2 m + 2 p − .Proof. a. For dimensional reasons, we must have P g m = αg m +2 p − plus possibly apower of g m , for some α ∈ F p , and P g m +2 p − = g m Y , for some polynomial Y . Theunstable condition requires that P m + p − g m +2 p − = g p m +2 p − , and, since m p , this equals, up to unit, P P m + p − g m +2 p − . For dimensional reasons, P m + p − g m +2 p − = βg m g p − m +2 p − + g m Z (3.10)for some β ∈ F p and some polynomial Z . By the Cartan formula (similar to (2.9)),the only way that P applied to (3.10) can yield g p m +2 p − is if both α and β are units.b. One way that the additional generators could affect the argument for part (a)would be if several of them (possibly the same one) were multiplied together to getinto the congruence of part (a). By the Cartan formula, P of such a product will stillinvolve some of these additional generators as factors, and so cannot yield the g p m +2 p − term on which the argument focuses. The other way, pointed out to the author byNishinobu, would be if there were a generator g d such that P g d = ug i m +2 p − for some u = 0 and i >
0, an eventuality excluded by our second hypothesis. If there weresuch a g d and also P m + p − g m +2 p − included a term g d g p − i m +2 p − , this would providean alternative way to achieve g p m +2 p − in P m + p − g m +2 p − . Lemma 3.11. If F [ g , g , g ] is an unstable A -algebra, then, mod decomposables, P g = u g and P g = u g with u i = 0 . The same conclusion holds for F [ g , g , g , g , g ] .Proof. The first part is a direct consequence of the second, so we just consider F [ g , g , g , g , g ]. We work mod the ideal ( g , g , g ). Then P g ≡ αg , P g ≡ βg , and P g ≡ γg , for some α , β , γ in F . This latter term complicatesthings somewhat. We also have P g ≡ P g ≡ P P g = ug with u = 0. We use the Cartanformula as in the previous proof. The only way to get to g by P is if β = 0,implying the result for P g . However, there are two ways that P P g might yield g , one via P ( g g ) andthe other via P ( g g ). Instead, we consider ( P ) P g . We must have P g ≡ δ g + δ g g for some δ i ∈ F . We compute( P ) g ≡ βγ g − β γ g g ( P ) ( g g ) ≡ β γ g + 5 β γ g g . Assuming that α = 0, so that the omitted terms of the form g Y in P g have( P ) (them) ≡
0, then we obtain ug ≡ ( P ) P ( g ) ≡ ( P ) ( δ g + δ g g ) ≡ δ ( βγ g − β γ g g ) + δ ( β γ g + 5 β γ g g ) . Here we have also used that omitted terms divisible by g or g have ( P ) (them) ≡ g imply β = 0, γ = 0, and some δ i = 0, but this then gives acontradiction regarding g g or g g . Thus the assumption that α = 0 must havebeen false. 4. In this section, we determine the v -periodic homotopy groups of the modular 5-compact groups X and X . We pass directly from invariant polynomials to Adamsoperations in K ∗ ( X ) and thence to v − π ∗ ( X ). In Theorems 4.15 and 4.20, we relatethe homotopy type of ( X ) and ( X ) to that of SU (20) and E . Theorem 4.15was conjectured by the author in an earlier version of this paper. It and Theorem4.20 and their proofs were provided to the author by John Harper.The input to determining the Adams module K ∗ ( X ; ˆ Z ) is the following resultdue to Aguad´e ([1]) and Maschke ([30]). Throughout the rest of the paper, we willdenote by m ( e ,... ,e k ) the smallest symmetric polynomial on variables x , . . . , x ℓ (thevalue of ℓ will be implicit) containing the term x e · · · x e k k . -COMPACT GROUPS 15 Theorem 4.1.
There is a reflection group G acting on ( ˆ Z ) , and there is a space BX and map BT → BX with BT = K (( ˆ Z ) , such that H ∗ ( BX ; ˆ Z ) ≈ H ∗ ( BT ; ˆ Z ) G , the invariants under the natural action of G on H ∗ ( BT ; ˆ Z ) = ˆ Z [ x , x , x , x ] with | x i | = 2 . Moreover, H ∗ ( BT ; ˆ Z ) G is a polynomial algebra on the following fourinvariant polynomials: f = m (4) − m (1 , , , f = m (8) + 14 m (4 , + 168 m (2 , , , f = m (12) − m (8 , + 330 m (4 , , + 792 m (6 , , , f = m (20) − m (16 , − m (12 , − m (14 , , , + 716 m (12 , , +1038 m (8 , , + 7632 m (10 , , , + 129012 m (8 , , , + 106848 m (6 , , , . Proof.
The group G is the subgroup of GL ( C ,
4) generated by the following fourmatrices. These can be seen explicitly in [1].12 − − − − − − − − − − − − , i − i , , . Since i ∈ ˆ Z , these act on ( ˆ Z ) , and this induces an action on H ∗ ( K (( ˆ Z ) , ≈ ˆ Z [ x , x , x , x ] . The invariants of this action were determined by Maschke ([30]) to be the polynomialsstated in the theorem. Although he did not state them all explicitly, they can be easilygenerated by: (a) define φ , ψ i , and χ as on his page 501, then (b) define Φ , . . . , Φ ason his page 504, and finally (c) let f = − Φ and f = F , f = F , and f = F as on his page 505. See also [36, p.287] for a reference to this work.Actually, Maschke’s work and that of [36] involved finding generators for the com-plex invariant ring. To see that these integral polynomials generate the invariant ringover ˆ Z , one must show that they cannot be decomposed over Z /
5. For example,one must verify that f cannot be decomposed mod 5 as a linear combination of f f , f f , f f , f f , and f . The need to do this was pointed out to the au-thor by Kasper Andersen in a dramatic way, as will be described prior to 5.6. Theverification here was performed by Andersen using a Magma program.Aguad´e ([1]) constructed the 5-compact group (
BX, X ) corresponding to this mod-ular reflection group.The approach based on the following proposition benefits from a suggestion ofClarence Wilkerson.
Proposition 4.2.
Let ( BX, X ) be a p -compact group corresponding to a reflectiongroup G acting on BT = K (( ˆ Z p ) n , . Suppose H ∗ ( BT ; ˆ Q p ) G = ˆ Q p [ f , . . . , f k ] , where f i is a polynomial in y , . . . , y n with y j ∈ H ( BT ; ˆ Q p ) corresponding to the j th factor.Let K ∗ ( BT ; ˆ Q p ) = ˆ Q p [[ x , . . . , x n ]] with x i the class of H − in the i th factor, where H is the complex Hopf bundle. Let ℓ ( x ) = log(1 + x ) . Then K ∗ ( BX ; ˆ Z p ) ≈ ˆ Q p [[ f ( ℓ ( x ) , . . . , ℓ ( x n )) , . . . , f k ( ℓ ( x ) , . . . , ℓ ( x n ))]] ∩ ˆ Z p [[ x , . . . , x n ]] . Proof.
The Chern character K ∗ ( BT ; ˆ Q p ) ch −→ H ∗ ( BT ; ˆ Q p ) satisfies ch( ℓ ( x i )) = y i andhence, since ch is a ring homomorphism, ch( f j ( ℓ ( x ) , . . . , ℓ ( x n ))) = f j ( y , . . . , y n ).It commutes with the action of G , and hence sends invariants to invariants. Indeed K ∗ ( BT ; ˆ Q p ) G = ˆ Q p [[ f ( ℓ ( x ) , . . . , ℓ ( x n )) , . . . , f k ( ℓ ( x ) , . . . , ℓ ( x n ))]] . (4.3)The invariant ring in K ∗ ( BT ; ˆ Z p ) is just the intersection of (4.3) with ˆ Z p [[ x , . . . , x n ]].Finally we use a result of [26] that K ∗ ( BX ; ˆ Z p ) ≈ K ∗ ( BT ; ˆ Z p ) G .Thus with f , f , f , f as in 4.1, we wish to find algebraic combinations of f ( ℓ ( x ) , . . . , ℓ ( x )) , . . . , f ( ℓ ( x ) , . . . , ℓ ( x ))which have coefficients in ˆ Z . A theorem of [26] which states that for a p -compactgroup BX there is an isomorphism K ∗ ( BX ; ˆ Z p ) ≈ ˆ Z p [[ g , . . . , g k ]], and the collapsing,for dimensional reasons, of the Atiyah-Hirzebruch spectral sequence H ∗ ( BX ; K ∗ (pt; ˆ Z p )) ⇒ K ∗ ( BX ; ˆ Z p ) (4.4)implies that the generators g j can be chosen to be of the form f j ( x , . . . , x n ) modhigher degree polynomials. -COMPACT GROUPS 17 Finding these algebraic combinations can be facilitated by using the p -typical logseries ℓ p ( x ) = X i ≥ x p n /p n . By [24], there is a series h ( x ) ∈ Z ( p ) [[ x ]] such that ℓ ( h ( x )) = ℓ p ( x ) and h ( x ) ≡ x mod( x ). Let x ′ i = h ( x i ). For any c e ∈ ˆ Q p with e = ( e , e , e , e ), we have X c e f ( ℓ p ( x ) , . . . , ℓ p ( x )) e · · · f ( ℓ p ( x ) , . . . , ℓ p ( x )) e (4.5)= X c e f ( ℓ ( x ′ ) , . . . , ℓ ( x ′ )) e · · · f ( ℓ ( x ′ ) , . . . , ℓ ( x ′ )) e , (4.6)where the sums are taken over various e . We will find c e so that (4.5) is in ˆ Z p [[ x , . . . , x ]].Thus so is (4.6), and hence also P c e f ( ℓ ( x ) , . . . , ℓ ( x )) e · · · f ( ℓ ( x ) , . . . , ℓ ( x )) e ,since h ( x ) ∈ Z ( p ) [[ x ]].A Maple program, which will be described in the proof, was used to prove thefollowing result.
Theorem 4.7.
Let f , f , f , f be as in 4.1, and let F j = F j ( x , . . . , x ) = f j ( ℓ ( x ) , . . . , ℓ ( x )) . Then the following series are -integral through grading 20; i.e., their coefficients ofall monomials x e · · · x e with P e i ≤ are 5-integral. F − F − F − F − F F + F − F F − F F − F − F − F − F − F F − F F − F F ; F − F − F − F − F F ; F − F − F − F F . Proof.
As observed in the paragraph preceding the theorem, it suffices to show thatthe same is true for e F j = f j ( x + x , . . . , x + x ). The advantage of this isto decrease the number of terms which must be kept track of and looked at. Wework one grading at a time, expanding relevant products of F j ’s as combinations ofmonomial symmetric polynomials in the fixed grading, and then solving a system oflinear equations to find the combinations that work. We illustrate with the calculationfor modifications of F in gradings 8 and then 12. In grading 8, we have e F = m (8) − m (5 , , , e F = m (8) + 14 m (4 , + 168 m (2 , , , e F = m (8) − m (5 , , , + 2 m (4 , + 144 m (2 , , , . We wish to choose a and b so that, in grading 8, a e F + b e F ≡ e F mod integers. Thuswe must solve a system of mod 5 equations for a and b with augmented matrix − − . The solution is a = 1, b = 1 /
2. We could also have used b = 3 since we are workingmod 5.Let e F ′ = e F − e F − e F . In grading 12, we have e F ′ = − m (12) − m (8 , − m (6 , , , + m (9 , , , + m (5 , , , e F = m (12) − m (8 , + 330 m (4 , , + 792 m (6 , , , e F e F = m + 15 m (8 , + 42 m (4 , , + 168 m (6 , , , − m (9 , , , − m (5 , , , − m (3 , , , e F = m (12) + 3 m (8 , + 6 m (4 , , + 432 m (6 , , , − m (9 , , , − m (5 , , , − m (3 , , , . We wish to choose a , b , and c so that, in grading 12, a e F + b e F e F + c e F ≡ e F ′ mod integers. Thus we must solve a system of equations mod 25 whose augmentedmatrix is − −
33 15 3 − − − −
36 600 − −
72 120 − − . The solution is a = 16, b = 7, and c = − e F in grading 12, then for e F ′′ , e F ′ , and e F ingradings 16 and then 20.By the observation in the paragraph involving (4.4), the modified versions of F , F ,and F given in Theorem 4.7, and also F , can be modified similarly in all subsequent -COMPACT GROUPS 19 gradings, yielding generators of the power series algebra K ∗ ( BX ; ˆ Z ) which we willcall G , G , G , and G . By [26], K ∗ ( X ; ˆ Z ) is an exterior algebra on classes z , z , z , and z in K ( − ) obtained using the map e : Σ X = ΣΩ BX → BX and Bottperiodicity B : K ( X ) → K − ( X ) by z i = B − e ∗ ( G i +1 ). The following determinationof the Adams operations is essential for our work on v -periodic homotopy groups.Here and elsewhere QK ( − ) denotes the indecomposable quotient. Theorem 4.8.
The Adams operation ψ k in QK ( X ; ˆ Z ) on the generators z , z , z , and z is given by the matrix k k − k k k − k − k
11 85 k − k k k − k − k − k
19 1225 k − k − k
19 15 k − k k . Proof.
We first note that ψ k ( ℓ ( x )) = ℓ ( ψ k x ) = ℓ (( x +1) k −
1) = log(( x +1) k ) = k log( x +1) = kℓ ( x ) . Since F j is homogeneous of degree 4 j in ℓ ( x i ), ψ k ( F j ) = k j F j . We can use thisto determine ψ k on the generators G i which are defined as algebraic combinations of F j ’s. We then apply e ∗ to this formula to obtain ψ k in K − ( X ; ˆ Z ). Since e ∗ annihi-lates decomposables, we need consider only the linear terms in the expressions whichexpress G i in terms of F j ’s. On the basis (over ˆ Q ) h e ∗ ( F ) , e ∗ ( F ) , e ∗ ( F ) , e ∗ ( F ) i ,the matrix of ψ k is D = diag( k , k , k , k ). On the basis (over ˆ Z ) h e ∗ ( G ) , e ∗ ( G ) , e ∗ ( G ) , e ∗ ( G ) i , it is P − DP , where P = − − − − − − is the change-of-basis matrix, obtained using the linear terms in 4.7. The matrix inthe statement of the theorem is obtained by dividing P − DP by k , since ψ k in K ( − )corresponds to ψ k /k in K − ( − ). We can use Theorem 4.8 to obtain the v -periodic homotopy groups of ( X ) asfollows. Theorem 4.9.
The groups v − π ∗ ( X ) (5) are given by v − π t − ( X ) ≈ v − π t ( X ) ≈ t Z / t ≡ ,
15 (20) Z / min(8 , ν ( t − − · )) t ≡ Z / min(12 , ν ( t − − · )) t ≡
11 (20) Z / min(20 , ν ( t − − · )) t ≡
19 (20) . Proof.
We use the result of [9] that v − π t ( X ) (5) is presented by the matrix (Ψ ) T (Ψ ) T − t I ! .We form this matrix by letting k = 5 and 2 in the matrix of Theorem 4.8 and letting x = 2 t , obtaining − − − − − − − x − − − − x − − − x − − x . (4.10)Pivoting on the units (over Z (5) ) in positions (5,2) and (7,4) and removing their rowsand columns does not change the group presented. We now have a 6-by-2 matrix,whose nonzero entries are polynomials in x of degree 1 or 2. If x t (cid:16) u B u (cid:17) with u i units, and sothe group presented is 0.Henceforth, we assume x ≡ x , and so we pivot on it, and remove its row and column.The five remaining entries are ratios of polynomials with denominator nonzero mod5. Let p , . . . , p denote the polynomials in the numerators. The group v − π t ( X ) (5) is Z / e , where e = min( ν ( p ( x )) , . . . , ν ( p ( x ))), where x = 2 t . We abbreviate ν ( − ) -COMPACT GROUPS 21 to ν ( − ) throughout the remainder of this section. We have p = − x − x + 33908441866 x p = − x − x + 31789306250 x p = − · + 11101145 · x − x + 39736328125 x p = 4 · − · x + 41656494140625000 x − x p = 1099511627776 − x + 1145324544 x − x + x . For values of m listed in the table, we compute and present in Table 4.12 the tuples( e , e , e , e ) so that, up to units, p i (2 m + y ) = 5 e + 5 e y + 5 e y + 5 e y (4.11)(plus y if i = 5). Considerable preliminary calculation underlies the choice of thesevalues of m . Table 4.12.
Exponents of polynomials im , , , ∞ , , , ∞ , , , ∞ , , , ∞ , , ,
115 3 , , , , , , , , ,
10 22 , , ,
19 4 , , ,
27 + 4 · , , , , , , , , ,
10 26 , , ,
19 8 , , ,
111 + 4 · , , , , , , , , , ∞ , , ,
19 12 , , ,
119 + 12 · , , , , , , , , ,
10 22 , , ,
19 20 , , , ν (2 · i −
1) = i + 1, as is easily proved by induction. Thus p (2 m +20 j ) = p (2 m + 2 m (2 j − p (2 m + 25 j · u ) , (4.13)with u a unit. Hencemin { ν ( p i (2 j )) : 1 ≤ i ≤ } = 3since p (2 j ) = p (2 + 5 ju ) = 5 + 5 · ju + 5(5 ju ) + (5 ju ) , omitting some unit coefficients. Here we have set y = 5 ju in (4.11). Replacing 3 by15 yields an identical argument. This yields the second line of Theorem 4.9. We use Table 4.12 to showmin { ν ( p i (2 · +20 j )) : 1 ≤ i ≤ } = min(20 , ν ( j )) = min(20 , ν (20 j )) . (4.14)Indeed, for ν ( j ) ≤
16, the minimum is achieved when i = 1, with the 4 coming as2 + 2 with one 2 being from the 25 in (4.13) and the other 2 being the first 2 in thelast row of Table 4.12. If ν ( j ) >
16, the minimum is achieved when i = 2, usingthe first 20 in the last row of 4.12. The last case of Theorem 4.9 follows easily from(4.14), and the other two parts of 4.9 are obtained similarly.To see that v − π t − ( X ) ≈ v − π t ( X ), we argue in three steps. First, the twogroups have the same order using [9, 8.5] and the fact that the kernel and cokernelof an endomorphism of a finite group have equal orders. Second, by [16, 4.4], apresentation of v − π t − ( X ) is given by Ψ Ψ − t ! , i.e. like that for v − π t ( X )except that the two submatrices are not transposed. Third, we pivot on this matrix,which is (4.10) with the top and bottom transposed, and find that we can pivot onunits three times, so that the group presented is cyclic.One of the factors in the product decomposition of SU (20) given in [31] is an H -space B (5) whose F -cohomology is an exterior algebra on classes of grading 7,15, 23, 31, and 39, and which is built from spheres of these dimensions by fibrations.By [40], there is a product decomposition( SU (20) /SU (15)) ≃ S × S × S × S × S . Let B (7 , , ,
39) denote the fiber of the composite B (5) → SU (20) → ( SU (20) /SU (15)) ρ −→ ( S ) . Theorem 4.15. (Harper) There is a homotopy equivalence ( X ) ≃ B (7 , , , . Note that this result requires more than 3.7 because the ranks of these H -spacesare not less than p −
1. We will provide Harper’s proof of this result in Section 6. Herewe just remark that our work above is required in the proof, for the entry in position(4 ,
3) of the matrix of 4.8 implies that the 39-cell of ( X ) is attached to the 23-cell -COMPACT GROUPS 23 by α , which is not detected by primary Steenrod operations. This information isrequired in order to compare the two spaces in 4.15.We can determine the Adams operations and v -periodic homotopy groups of ( X ) by an argument very similar to that used above for ( X ) . We shall merely sketch.The analogue of Theorem 4.1 is Theorem 4.16.
There is an isomorphism H ∗ ( BX ; ˆ Z ) ≈ H ∗ ( BT ; ˆ Z ) G , where G has the four generators given for G in the proof of 4.1 and also (cid:18) − (cid:19) .Then H ∗ ( BT ; ˆ Z ) G is a polynomial ring on the generators f , f , and f given in4.1 together with f = m (24) − m (20 , + 1023 m (16 , + 2180 m (12 , + 1293156 m (8 , , , +267096 m (12 , , , + 2121984 m (6 , , , + 620352 m (10 , , , − m (14 , , , − m (10 , , , − m (12 , , − m (16 , , − m (8 , , − m (18 , , , . The analogue of Theorem 4.7 is
Theorem 4.17.
Let f , f , f , f be as in 4.16, and let F j = f j ( ℓ ( x ) , . . . , ℓ ( x )) . Then the following series are 5-integral through grading 24. F − F − F − F − F F − F − F − F F − F − F − F F − F − F − F F − F − F . The analogue of 4.8 is
Theorem 4.18.
The Adams operation ψ k in K ( X ; ˆ Z ) on the generators z , z , z , and z is given by the matrix k k − k k k − k − k
19 15 k − k k k − k − k − k
23 2125 k − k − k
23 35 k − k k . The analogue of Theorem 4.9 is
Theorem 4.19.
The groups v − π ∗ ( X ) (5) are given by v − π t − ( X ) ≈ v − π t ( X ) ≈ t Z / t ≡ ,
15 (20) Z / min(8 , ν ( t − − · )) t ≡
11 (20) Z / min(12 , ν ( t − − · )) t ≡
19 (20) Z / min(20 , ν ( t − − · )) t ≡
23 (20) . In [39, Proposition 2.3], it is proved that the exceptional Lie group E , localizedat 5, admits a product decomposition as X ( E ) × X ( E ) with H ∗ ( X ( E ); F ) ≈ Λ( x , x , x , x ). In Section 6, we will prove the following result, which was pointedout by John Harper. Theorem 4.20. (Harper) There is a homotopy equivalence ( X ) ≃ X ( E ) . The 7-primary modular case
In this section, we first give in Theorem 5.1 new explicit formulas for the six poly-nomials which generate as a polynomial algebra the invariant ring of the complexreflection group G of [36], called the Mitchell group in [14]. Over ˆ Z , the invariantring of G is also a polynomial algebra, but the generators must be altered slightlyfrom the complex case, as we show prior to 5.6. Next we use this information to findexplicit generators for K ∗ ( BX ; ˆ Z ) in 5.6, and from this the Adams operations in QK ( X ; ˆ Z ) in 5.15. These in turn enable us to compute the v -periodic homotopygroups v − π ∗ ( X ) (7) . Finally, we prove in 5.17 that ( X ) has the homotopy type ofa space formed from SU (42). This result was conjectured by the author and provedby John Harper. Theorem 5.1.
The complex invariants of the reflection group G (defined in theproof ) form a polynomial algebra C [ x , . . . , x ] G ≈ C [ f , f , f , f , f , f ] -COMPACT GROUPS 25 with generators given by f k = (1+( − k k − · m (6 k ) + k X s =1 (cid:16) k s (cid:17) (1+( − k + s k − ) m (6 k − s, s ) + X e ( e ) m e , where e ranges over all partitions e = ( e , . . . , e r ) of k with ≤ r ≤ satisfying e i ≡ e j mod 3 for all i, j , and e i ≡ mod 3 if r < . Here also ( e ) denotes the multinomialcoefficient ( e + · · · + e r )! / ( e ! · · · e r !) , and m e the monomial symmetric polynomial,which is the shortest symmetric polynomial in x , . . . , x containing x e · · · x e r r . For example, we have • f = − m (6) + 40 m , + 720 m (1 , , , , , ; • f = 136 m (12) − (cid:16) (cid:17) m (9 , + 28 (cid:16) (cid:17) m (6 , + P ( e ) m e , where e ranges over { (6 , , , (3 , , , , (2 , , , , , , (7 , , , , , , (4 , , , , , } . • f = (1 − · ) m (18) + (cid:16) (cid:17) (1+27 ) m (15 , + (cid:16) (cid:17) (1 − ) m (12 , + (cid:16) (cid:17) (1 + 27 ) m (9 , + P ( e ) m e , where e ranges over { (12 , , , (9 , , , (9 , , , , (6 , , , (6 , , , , (6 , , , , , (3 , , , , , , (13 , , , , , , (10 , , , , , , (7 , , , , , , (4 , , , , , , (7 , , , , , , (8 , , , , , , (5 , , , , , } Proof of Theorem 5.1.
As described in [36], the reflection group G is generated byreflections across the following hyperplanes in C : x i − x j = 0, x − ωx = 0, and x + x + x + x + x + x = 0. Here ω = e πi/ . It follows easily that G is generatedby all permutation matrices together with the following two: ω ω , I − (5.2)In [14], Conway and Sloane consider G instead as the automorphisms of a certain Z [ ω ]-lattice in C . The lattice has 756 vectors of norm 2. There are none of smallerpositive norm. 270 of these vectors are those with ω a in one position, − ω b in another, and 0 in the rest. Here, of course, a and b can be 0, 1, or 2. The other 486 are thoseof the form ± √− ( ω a , . . . , ω a ) such that P a i ≡ G is consistent with thereflection approach, one can verify that the reflection matrices permute these 756vectors. It is obvious that permutation matrices do, and easily verified for the firstmatrix of (5.2). The second matrix of (5.2), which has order 2, sends • ( ω, ω , , , ,
0) to √− ( ω , ω, , , , • √− (1 , , , , ,
1) to − √− (1 , , , , , • √− (1 , , , ω, ω, ω ) to − √− ( ω, ω, ω, , , • √− (1 , , ω, ω, ω , ω ) to itself.After permutation, negation, and multiplication by ω , this takes care of virtually allcases.Let p m ( x , . . . , x ) = X ( v ,... ,v ) ( v x + · · · + v x ) m , (5.3)where the sum is taken over the 756 vectors described above. Then p m is invariantunder G for every positive integer m . It is proved in [14, Thm.10] that the ring ofcomplex invariant polynomials is given by C [ x , . . . , x ] G = C [ p , p , p , p , p , p ] . (5.4)In [14], several other lattices isomorphic to the above one are described, any ofwhich can be used to give a different set of vectors v and invariant polynomials p m , still satisfying (5.4). The one that we have selected seems to give the simplestpolynomials; in particular, the only ones with integer coefficients.We have p k = S + S , where S = P i = j P a,b =0 ( ω a x i − ω b x j ) k , with 1 ≤ i, j ≤ S = 2( − k X a i =0 ( ω a x + · · · + ω a x + ω − a −···− a x ) k . The coefficient of 2 on S is due to the ±
1. Note that the sum for S has 6 · · terms, while that for S has 3 terms. Next note that if a term T k occurs in eithersum, then so does ( ωT ) k and ( ω T ) k , and all are equal. Thus we obtain S = -COMPACT GROUPS 27 P i = j P b =0 ( x i − ω b x j ) k and S = 3 2( − k X a ,... ,a =0 ( x + ω a x + · · · + ω a x + ω − a −···− a x ) k . We simplify S further as S = 3 k X ℓ =0 ( − ℓ (cid:16) kℓ (cid:17) X i = j x ℓi x k − ℓj X b =0 ω bℓ = 9 k X s =0 ( − s (cid:16) k s (cid:17) X i = j x si x k − sj = 18(5 m k + k X s =1 ( − s (cid:16) k s (cid:17) m (6 k − s, s ) ) . At the first step, we have used that P b =0 ω bℓ equals 0 if ℓ ℓ ≡ P i = j x si x k − sj equals m (6 k − s, s ) if s
6∈ { , k, k } , it equals 2 m (3 k, k ) if s = k , and equals 5 m (6 k ) if s = 0 or 2 k .The sum S becomes S = 6( − k X e ( e ) X a =0 ( ω e − e ) a · · · X a =0 ( ω e − e ) a x e · · · x e = 6( − k X e ≡···≡ e (3) ( e )3 x e · · · x e . Then ( − k ( S + S ) /
486 equals the expression which we have listed for f k in thestatement of the theorem. We have chosen to work with this rather than p k itself fornumerical simplicity. It is important that the omitted coefficient is not a multiple of7. For good measure, we show that (5.3) is 0 if m m T m by ( ωT ) m leaves the sums like S and S for (5.4) unchangedwhile, from a different perspective, it multiplies them by ω m . Thus the sums are 0.If m ≡ S corresponding to P x si x m − sj occurs with oppositesign to that corresponding to P x m − si x sj , and so S = 0. For S , the ( ± m will causepairs of terms to cancel. Remark 5.5.
The only other place known to the author where formulas other than(5.3) for these polynomials exist is [28], where they occupy 190 pages of dense textwhen printed.As pointed out by Kasper Andersen, f − ( f ) is divisible by 7. This is easily seenby expanding ( f ) = ( P ( v x + · · · + v x ) ) by the multinomial theorem. The needfor this became apparent to Andersen, as the author had thought that the invariantring of G over ˆ Z was ˆ Z [ f , . . . , f ], and this would have led to an impossibleconclusion for the Adams operations in QK ( X ; ˆ Z ).Let h = ( f − ( f ) ). Then we have the following result, for which we aregrateful to Andersen. Theorem 5.6.
The invariant ring of G over ˆ Z is given by ˆ Z [ x , . . . , x ] G = ˆ Z [ f , f , f , f , f , h ] . Proof. A Magma program written and run by Andersen showed that each of theseasserted generators is indecomposable over Z /
7. (This is what failed when f wasused; it equals ( f ) over Z / f is invariant under G , it follows from (5.4) that it can be decomposedover C in terms of f , f , f , f , and f . The nature of the coefficients in thisdecomposition was not so clear. It turned out that all coefficients were rationalnumbers which are 7-adic units. We make this precise in Theorem 5.7. f can be decomposed as q f f + q f f + q f + q f f + q f f f + q f + q f f + q f f + q f f + q f -COMPACT GROUPS 29 with q = 944610925401 / q = 733671261 / q = 243068633 / q = − / q = − / q = − / q = 4011206338081535787030788541 / q = 701461342458322269763709951654931 / q = − / q = 26589469730264682368719198549833 / Each of these coefficients q i is a 7-adic unit; i.e. no numerator or denominator isdivisible by 7.Proof. The ten products, f f , . . . , f , listed above are the only ones possible. Weexpress each of these products as a combination of monomial symmetric polynomials m e . We use Magma to do this. The length of m ( e ,... ,e r ) is defined to be r . We onlykept track of components of the products of length ≤
4. This meant that we onlyhad to include components of length ≤ f k being multiplied.There were 34 m e ’s of length ≤
4. These correspond to the partitions of 36 intomultiples of 3. (Note that monomials with subscripts ≡ Magma expressed each monomial such as f f or f as an integer combi-nation of these, plus monomials of greater length. We just ignored in the output allthose of greater length. The coefficients in these expressions were typically 12 to 15digits. We also wrote f as a combination of monomial symmetric polynomials oflength ≤
4, ignoring the longer ones. This did not require any fancy software, justthe multinomial coefficients from Theorem 5.1.
Now we have a linear system of 34 linear equations with integer coefficients in 10unknowns. The unknowns are the coefficients q i in the equation at the beginning of5.7, and the equations are the component monomials of length ≤
4. Miraculously,there was a unique rational solution, as given in the statement of this theorem.If it were not for the fact that the Conway-Sloane theorem 5.4 guarantees that theremust be a solution when all monomial components (of length ≤
6) are considered,then we would have to consider them all, but the fact that we got a unique solutionlooking at only the monomial components of length ≤ QK ( X ; ˆ Z ).Similarly to 4.7, we let ℓ ( x ) = ln(1 + x ), and F i = F i ( x , . . . , x ) = f i ( ℓ ( x ) , . . . , f ( ℓ ( x ))) . (5.8)A major calculation is required to modify the classes F i so that their coefficientsare in ˆ Z ; i.e. they do not have 7’s in the denominators. As observed after (4.4), itwill be enough to accomplish this through grading 42 (with grading of x i consideredto be 1). Theorem 5.9.
The following expressions are 7-integral through grading 42: • F + F + F ; • F + F + F + F ; • F + F + F + F + F ; • F + F + F + F + F + F ; • F + F + F + F + F + F + F . It was very surprising that just linear terms were needed here. Decomposableterms were certainly expected. The analogue for G in 4.7 involved many decom-posables. It would be interesting to know why Theorem 5.9 works with just linearterms; presumably this pattern will continue into higher gradings. Proof of Theorem 5.9.
Similarly to the proof of 4.7, we define e F i = e F i ( x , . . . , x ) = f i ( ℓ p ( x ) , . . . , ℓ p ( x )) , -COMPACT GROUPS 31 and observe that a polynomial in the e F i ’s is 7-integral if and only if the same poly-nomial in the F i ’s is.Next note that in the range of concern for Theorem 5.9 ℓ ( x ) = x + x /
7. If wedefine h i = h i ( x , . . . , x ) = f i ( x + x , . . . , x + x ) , then 5.9 is clearly equivalent to Statement 5.10.
For t ≥ and grading ≤ , • h + 5 h + 22 h ≡ mod t in grading
30 + 6 t ; • h + 4 h + 45 h + 104 h ≡ mod t in grading
24 + 6 t ; • h + 3 h + 20 h + 157 h + 526 h ≡ mod t in grading
18 + 6 t ; • h + 2 h + 45 h + 109 h + 1391 h + 6201 h ≡ mod t ingrading
12 + 6 t ; • h + h + 22 h + 204 h + 1107 h + 9682 h + 100682 h ≡ mod t in grading t . We use
Maple to verify 5.10. Our f i ’s are given in Theorem 5.1 in terms of m e ’s. Toevaluate m e ( x + x , . . . , x + x ), the following result keeps the calculation manageable(e.g. it does not involve a sum over all permutations). Partitions can be writteneither in increasing order or decreasing order; we use increasing. If ( a , . . . , a r ) isan r -tuple of positive integers, let s ( a , . . . , a r ) denote the sorted form of the tuple;i.e. the rearranged version of the tuple so as to be in increasing order. For example, s (4 , , ,
2) = (2 , , , Proposition 5.11.
The component of m ( e ,... ,e r ) ( x + x , . . . , x + x ) in grading P e i + 6 t is X j P ( e + 6 j , . . . , e r + 6 j r ) P ( e , . . . , e r ) e j ! · · · e r j r ! m s ( e +6 j ,... ,e r +6 j r ) , where j = ( j , . . . , j r ) ranges over all r -tuples of nonnegative integers summing to t ,and P ( a , . . . , a r ) is the product of the factorials of repetend sizes. For example, P (4 , , ,
3) = 2! because there are two 3’s, P (3 , , , , ,
2) = 3!2!,and P (3 , , ,
1) = 1.
Example 5.12.
We consider as a typical example, the component of m (3 , , , ( x + x , . . . , x + x ) in grading 42. Table 5.13 lists the possible values of j and the contribution to the sum.The final answer is the sum of everything in the right hand column. Table 5.13.
Terms for Example 5.12 j term (2 , , , (cid:16) (cid:17) m , , , (0 , , , (cid:16) (cid:17) m , , , (0 , , , (cid:16) (cid:17) m , , , (0 , , , (cid:16) (cid:17) m , , , (1 , , ,
0) 3 · · m , , , (1 , , ,
0) 3 · m , , , (1 , , ,
1) 3 · m , , , (0 , , ,
0) 3 · m , , , (0 , , ,
1) 3 · m , , , (0 , , ,
1) 9 · m , , , Proof of Proposition 5.11. m ( e ,... ,e r ) ( x + x , . . . , x + x ) is related to X σ ( x e σ (1) + (cid:16) e (cid:17) x e +6 σ (1) + (cid:16) e (cid:17) x e +12 σ (1) + · · · ) · · · ( x e r σ ( r ) + (cid:16) e r (cid:17) x e r +6 σ ( r ) + · · · )(5.14)summed over all permutations σ in Σ r . If t values of e i are equal, then (5.14) willgive t ! times the correct answer. That is the reason that we divide by P ( e ). If( e +6 j , . . . , e r +6 j r ) contains s equal numbers, then the associated m will be obtainedfrom each of s ! permutations, which is the reason that P ( e +6 j , . . . , e r +6 j r ) appearsin the numerator.At first, mimicking 4.7, we were allowing for products of h ’s in addition to thelinear terms which appear in 5.10, but it was turning out that what was needed tosatisfy the congruences was just the linear term. If just a linear term was going towork, the coefficients could be obtained by just looking at monomials of length 1.They were computed by Maple , using that, by 5.1 and 5.11, the coefficient of m (6 k +6 t ) in h k is (1 + ( − k k − · (cid:16) kt (cid:17) . Write the k th expression from the bottom of 5.10 -COMPACT GROUPS 33 as P j ≥ a j,k h k +6 j . We require that the coefficient of m (6 k +6 t ) in P tj =0 a j,k h k +6 j is 0mod 7 t . But this coefficient equals t X j =0 a j,k (cid:16) k +6 jt − j (cid:17) (1 + ( − k + j k + j − · . We solve iteratively for a j,k , starting with a ,k = 1, and obtain the values in 5.10.Note that it first gives a , ≡ − a , would be different than 22. So these numbers a j,k are not uniquely determined. These different choices just amount to choosing adifferent basis for QK ( X ; ˆ Z ).Verifying Statement 5.10 required running many Maple programs. For each line of5.10, a verification had to be made for each relevant t -value, from two t -values forthe first line down to six t -values for the last line. Moreover, for each of these pairs(line number, t -value), it was convenient to use a separate program for monomialsof each length 2, 3, 4, and 5, and then, for monomials of length 6, it was doneseparately for those with subscripts congruent to 0, 1, or 2 mod 3. Thus altogether(2 + 3 + 4 + 5 + 6)(4 + 3) = 140 Maple programs were run. The programs had enoughsimilarity that one could be morphed into another quite easily, and a more skillfulprogrammer could incorporate them all into the same program.Note that expanding from f j to h j does not change the number of componentsin monomials, nor does it change the mod 3 value of the sum of the subscripts (i.e.exponents) in the monomials. This is simpler than the situation in the proof of 5.7.The algorithm is quite easy. For each combination of h ’s in 5.10, replace each h j bythe combination of m e ’s in f j in 5.1, but expanded using 5.11.To obtain the Adams operations in QK ( X ; ˆ Z ), we argue similarly to the para-graph which precedes Theorem 4.8. First note that F decomposes in terms of F i ’sexactly as does f in terms of f i ’s in 5.7. We can modify by decomposables indimensions greater than 42 to obtain 7-integral classes G , G , G , G , and G which agree with the classes of 5.9 (with F decomposed) through dimension 42.There is also a 7-integral class G which agrees with ( F − ( F ) ) in dimension 42.These generate K ∗ ( BX ; ˆ Z ) as a power series algebra. As in the preamble to 4.8, But we need not bother to do so explicitly. then z i := B − e ∗ ( G i +1 ) for i = 5, 11, 17, 23, 29, and 41 form a basis for QK ( X ; ˆ Z ),and e ∗ annihilates decomposables.Similarly to the situation for ( X ) in the proof of 4.8, if we let P = , then the matrix of ψ k on the basis { z , z , z , z , z , z } is P − diag( k , k , k , k , k , k ) P. The entries in the last row of P are 7 times the coefficients of F in 5.9 reduced mod1. Those coefficients were multiplied by 7 because z is related to F rather thanto F .Using this, we compute the v -periodic homotopy groups, similarly to 4.9. Note theremarkable similarity with that result. Here, of course, ν ( − ) denotes the exponent of7 in an integer. Theorem 5.15.
The groups v − π ∗ ( X ) (7) are given by v − π t − ( X ) ≈ v − π t ( X ) ≈ t Z / t ≡ ,
35 (42) Z / min(12 , ν ( t − − · )) t ≡
11 (42) Z / min(18 , ν ( t − − · )) t ≡
17 (42) Z / min(24 , ν ( t − − · )) t ≡
23 (42) Z / min(30 , ν ( t − − · )) t ≡
29 (42) Z / min(42 , ν ( t − − · )) t ≡
41 (42) . Proof.
The group v − π t ( X ) (7) is presented by ( ψ ) T ( ψ ) T − t I ! , since 3 generates Z / × .We let x = 3 t and form this matrix analogously to (4.10). Five times we can pivot onunits, removing their rows and columns, leaving a column matrix with 7 polynomialsin x . The 7-exponent of v − π t ( X ) (7) is the smallest of that of these polynomials(with x = 3 t ). This will be 0 unless x ≡ t ≡ -COMPACT GROUPS 35 mod 6. We find that two of these polynomials will always yield, between them, thesmallest exponent. Similarly to (4.11) and Table 4.12, we write these polynomials as p i (3 m + y ) for carefully-chosen values of m . Much preliminary work is required todiscover these values of m . Ignoring unit coefficients and ignoring higher-power termswhose coefficients will be sufficiently divisible that they will not affect the divisibility,these polynomials will be as in Table 5.16. Table 5.16.
Certain p i (3 m + y ) , (linear part only) m p p ,
35 7 + 7 y ≥ + 7 y
11 + 12 · + 7 y + 7 y
17 + 18 · + 7 y + 7 y
23 + 18 · + 7 y + 7 y
29 + 12 · + 7 y + 7 y
41 + 24 · + 7 y + 7 y The claim of the theorem follows from Table 5.16 by the same argument as wasused in the proof of 4.9. For t in the specified congruence, if 3 t = 3 m + y , then ν ( y ) = ν ( t − m ) + 1 ≥
2, similarly to (4.13). For example, if t ≡
11 mod 42, and3 t = 3 · + y , then ν ( y ) = ν ( t − − · ) + 1. Thus min( ν ( p (3 t )) , ν ( p (3 t )))will be determined by the 7 in p if t ≡ ,
35 (42), while in the other cases, it isdetermined by the 7 y in p or the constant term in p .The groups v − π t − ( X ) are cyclic by an argument similar to the one described atthe end of the proof of 4.9, and have the same order as v − π t ( X ) for the standardreason described there.Similarly to the discussion preceding Theorem 4.15, one of the factors in the productdecomposition of SU (42) given in [31] is an H -space B (7) whose F -cohomology isan exterior algebra on classes of grading 11, 23, 35, 47, 59, 71, and 83, and whichis built from spheres of these dimensions by fibrations. Using [40], we can obtain adegree-1 map B (7) → S . Let B := B (11 , , , , ,
83) denote its fiber. Thefollowing result was conjectured by the author and proved by John Harper. Its proofwill be described in the last line of the paper.
Theorem 5.17. (Harper) There is a homotopy equivalence ( X ) ≃ B . Proofs provided by John Harper
In this section, we provide proofs of Theorems 4.15, 4.20, and 5.17, which wereexplained to the author by John Harper. We begin with the following strengtheningof Theorem 3.7.
Theorem 6.1.
Theorem 3.7 is true for r ≤ p − .Proof. Although the proof presented in [15] works for r = p − r < p −
1, we take this opportunity to explain some aspects of it more thoroughly. For λ = 0 ∈ Z /p , a π λ -space (resp. Q λ -space) is one which admits a self-map inducingmultiplication by λ in π ∗ ( − ) ⊗ Z /p (resp. QH ∗ ( − ; Z /p )). A π λ -map (resp. Q λ -map)is a map between spaces of the indicated type which commutes up to homotopy withthe self-maps.The first part of the proof, extending [15, Thm 1.3], involves showing that if X isan H -space of rank r ≤ p − π λ -map X → S n whose homotopy fiber Y is an H -space of rank r − r = p − r < p −
1. That Y is an H -space follows from[15, Thm 1.1] since its rank is less than p −
1. This construction can be iterated toyield (3.8).We use Lemma 6.2 to deduce the part of 6.1 which says that the homotopy type of X is determined by certain homotopy classes. Suppose we have (3.8) and a primedversion. Suppose we have shown that X i and X ′ i are p -equivalent H -spaces and thatthe elements α ∈ π n i +1 − ( X i ) and α ′ ∈ π n i +1 − ( X ′ i ) correspond under this equivalence.Then, in the notation of 6.2, there are equivalences as π λ -spaces X i +1 ≃ ( X i ) α ≃ ( X ′ i ) α ′ ≃ X ′ i +1 . If i + 1 < p −
1, then by [15, Thm 1.1], there are p -equivalent H -space structureson X i +1 and X ′ i +1 , extending the induction. Note that for X p − and X ′ p − we do notassert an equivalence as H -spaces, only as π λ -spaces. Lemma 6.2.
Suppose Y → X g −→ S n is a fibration, with X a π λ -space and Y an H -space with rank ( Y ) < p − . Let α ∈ π n − ( Y ) denote ∂ ( ι n ) in the homotopy sequence of -COMPACT GROUPS 37 the fibration. As in [15, p.351] , let Y α = D n + × Y ∪ c D n − × Y , where c ( x, y ) = ( x, α ( x ) y ) ,using the H -space multiplication of Y . Then there is a homotopy equivalence X ≃ Y α which is a π λ -map.Proof. We expand on some aspects of the proof given in [15, p.358] and correct severalconfusing typos. As explained there, Y α is a π λ -space and admits an inclusion Y ∪ α e n ֒ → Y α .On the other hand, the given fibration is fiber homotopy equivalent to Y → X ′ → S n , with X ′ = D n + × Y ∪ γ D n − × Y , where γ : S n − × Y → S n − × Y is the clutchingfunction defined by the homotopy equivalences D n ± × Y → g − ( D n ± ). We will workwith X ′ , but in the end may replace it with X . Note that X ′ inherits the π λ -structureof X .Since γ | S n − × {∗} represents α , the relative n -skeleta ( Y α , Y ) n and ( X ′ , Y ) n bothequal Y ∪ α e n . Thus the n th stages of the Postnikov systems of Y α and X ′ are builtfrom the same Eilenberg-MacLane spaces and same k -invariants, and so there existsa π λ -map ( Y α ) n → ( X ′ ) n of these n th stages, inducing a cohomology isomorphismin dimension ≤ n . We wish to show that this extends to a π λ -map Y α → X ′ , whichwill then be a homotopy equivalence by the Five Lemma applied to the homotopysequences of the fibrations Y → Y α → S n and Y → X ′ → S n , and Whitehead’sTheorem.This requires that we consider the primitive Postnikov systems (PPS) of the map X ′ → ( X ′ ) n . The primitive Postnikov system , as described, for example, in [27,p.426ff], applies to a rational equivalence, and gives a Postnikov tower in which allfibers are mod p Eilenberg-MacLane spaces. Our map X ′ → ( X ′ ) n is a rationalequivalence because the only infinite homotopy groups of the spheres that build X ′ are present in ( X ′ ) n . By [15, Lemma 2.8], the maps in the PPS are π λ -maps. We willalso use that, by [15, Lemma 2.4], all our π λ -maps are also Q λ -maps.We will show that the composite Y α → ( Y α ) n → ( X ′ ) n lifts through the PPS of X ′ → ( X ′ ) n . Assume there is a lifting to a π λ -map Y α → E s , where E s is some stagein the PPS. Since the k -invariants in the PPS are λ -eigenvectors by [15, Lemma 2.8],so are their images in H ∗ ( Y α ; F p ). A basis for H ∗ ( Y α ; F p ) consists of products ofno more than p − λ -eigenvector for the Q λ -map of Y α . An element in this basis which is not one of the generators is a λ i -eigenvector for some 2 ≤ i ≤ p −
1. Since, for such i , λ i λ mod p , we deduce that a nonzeroimage of a k -invariant can only equal one of the generators of H ∗ ( Y α ; F p ). However,the k -invariants are in dimension greater than that of the generators of H ∗ ( Y α ; F p ).We conclude that the image of the k -invariants must equal 0, and so the map liftsto Y α → E s +1 . By [15, 2.7], this lifting may be chosen to be a π λ -map. Since thePPS has only finitely many stages through dim( Y α ), we obtain the desired lifting Y α → X ′ .We now apply these general results to our specific situations. Proof of Theorem 4.20.
Both spaces ( X ) and X ( E ) are H -spaces with mod-5 co-homology an exterior algebra on classes of dimension 15, 23, 39, and 47. By Theorem6.1, there exist diagrams of fibrations S −−−→ X −−−→ X −−−→ ( X ) f y f y f y S S S and S −−−→ X ′ −−−→ X ′ −−−→ X ( E ) f ′ y f ′ y f ′ y S S S , and the homotopy types of the spaces are determined by relevant elements of homo-topy groups.Always localized at 5, we have π ( S ) ≈ Z / α . The elements in π ( S ) which determine both X and X ′ are nonzero multiples of α , and so by3.7 we obtain an equivalence X ′ ≃ X . The claim about these homotopy classes isproven for ( X ) from the entry in position (2,1) in the Adams operation matrixin 4.18; Adams’ e -invariant says that α attaching maps are present iff the relevantAdams operation involves u ( k n − k n + p − ) /p with u a unit. Similarly for X ( E ), thepresence of α can be deduced from the Adams operations in E as given in [17, Prop3.5], although here the well-known Steenrod algebra action in H ∗ ( E ) also impliesthe attaching map. We now identify X ′ with X in our notation. or a linear combination of generators in the same dimension -COMPACT GROUPS 39 The homomorphism π ( X ) → π ( S ) is a surjection Z / → Z /
5. See, e.g.,Diagram 3.5. The ( k − k ) in position (3,2) of 4.18 and in the formula for ψ k ( x )in [17, Prop 3.5] tell that in both of our sequences, the determining element of π ( X )is a generator, i.e., it maps to α ∈ π ( S ), and so, by Theorem 3.7, there is ahomotopy equivalence X ≃ X ′ . This is probably the only place that any of ourcalculations with ( X ) are required in proving 4.20. The α attaching maps can beseen by Steenrod operations, but α requires secondary operations, which were usedin [23] to see the α attaching map in X ( E ), or Adams operations in K -theory, aswe have done.We identify X and X ′ . The homomorphism π ( X ) → π ( S ) is a surjection Z / → Z /
5. (The cyclicity of π ( X ) can be proved easily using [17, Prop 5.5].)As in the previous two paragraphs, Adams operations imply that our determiningelement in both sequences is a generator, i.e., it maps to α ∈ π ( S ), and so weobtain the asserted equivalence from Theorem 6.1. Proof of Theorem 4.15.
We first show that the map B → S described prior to 4.15can be chosen to be a π λ -map. By [15, Lemma 2.8], the generating map S → Z ( Z ,
31) has a PPT in which all maps are π λ -maps. The map B g −→ K ( Z ,
31) is a π λ -map, since π ( B ) is cyclic. By [40], g lifts to a map B → S . By the LiftingTheorem, [15, 2.7a], this lifting, through each stage of the PPT, is a π λ -map.Thus, by [15, Lemma 2.3], the space which we call B (7 , , ,
39) in 4.15 is a π λ -space. Let f denote the composite B (7 , , , → B → SU (20) → S . Its fiber, B = B (7 , , SU (12) , and hence is an H -space.On the other hand, Theorem 6.1 yields fibrations of H -spaces, localized at 5, S −−−→ X −−−→ X −−−→ ( X ) y y y S S S (6.3)We can almost apply Theorem 6.1 to obtain the desired equivalence of ( X ) and B (7 , , , B (7 , , , H -space. Instead, we first use Theorem 3.7 to show that X (of (6.3)) and B are homo-topy equivalent H -spaces. Note that in the proof of 6.1 we obtained equivalent H -structures as long as the rank was less than p −
1. In this case, the homotopy classesthat must correspond are, in both cases, α ∈ π ( S ) (5) ≈ Z / π ( B (7 , ≈ Z /
25, mapping to α ∈ π ( S ). The α ’s in B are well-known bythe action of P , while in X they may be deduced from entries in the matrix of 4.8in positions (2 ,
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