ODD-PRIMARY HOMOTOPY EXPONENTS OF COMPACT
SIMPLE LIE GROUPS
DONALD M. DAVIS AND STEPHEN D. THERIAULT
Abstract.We note that a recent result of the second author
yields upper bounds for odd-primary homotopy exponents of com-
pact simple Lie groups which are often quite close to the lower
bounds obtained from v1-periodic homotopy theory.
1.Statement of results
The homotopy p-exponent of a topological space X, denoted expp(X), is the lar*
*gest
e such that some homotopy group ssi(X) contains a Z=pe-summand.1 In work dat-
ing back to 1989, the first author and collaborators have obtained lower bounds*
* for
expp(X) for all compact simple Lie groups X and all primes p by using v1-period*
*ic
homotopy theory. Recently, the second author ([11]) proved a general result, st*
*ated
here as Lemma 2.1, which can yield upper bounds for homotopy exponents of spaces
which map to a sphere. In this paper, we show that these two bounds often lead *
*to
a quite narrow range of values for expp(X) when p is odd and X is a compact sim*
*ple
Lie group.
Our first new result, which will be proved in Section 2, combines Lemma 2.1 w*
*ith
a classical result of Borel-Hirzebruch.
Theorem 1.1. Let p be odd.
a. If n < p2+ p, then expp(SU(n)) n - 1 + p((n - 1)!).
ibn-2_c-p+2j
b. If n p2+ 1, then expp(SU(n)) n + p - 3 + p-12 .
Here and throughout, p(-) denotes the exponent of p in an integer, p is an o*
*dd
prime, and bxc denotes the integer part of x . All spaces are localized at p. *
* It is
__________
Date: January 17, 2006.
Key words and phrases. Homotopy group, Lie group.
2000 Mathematics Subject Classification: 57T20, 55Q52.
1Some authors (e.g. [11]) say that peis the homotopy p-exponent.
1
2 DONALD M. DAVIS AND STEPHEN D. THERIAULT
useful to note the elementary fact that
p(m!) = bm_pc + b_m_p2c + . .,.
and the well-known fact that p(m!) bm-1_p-1c.
Theorem 1.1(a) compares nicely with the following known result.
Theorem 1.2. a. ([7, 1.1]) For any prime p, expp(SU(n)) n - 1 + p(bn_pc!).
b. ([8, 1.8]) If p is odd, 1 t < p, and tp-t+2 n tp+1, then expp(SU(n))*
* n.
Thus we have the following corollary, which gives the only values of n > p in*
* which
the precise value of expp(SU(n)) is known.
Corollary 1.3. If p is an odd prime, and n = p+1 or n = 2p, then expp(SU(n)) = *
*n.
When n = p+1, this was known (although perhaps never published) since, locali*
*zed
at p, SU(p + 1) ' B(3, 2p + 1) x S5x . .x.S2p-1, the exponent of which follows *
*from
Proposition 1.4 together with the result of Cohen, Moore, and Neisendorfer ([5]*
*) that
if p is odd, then expp(S2n+1) = n. Here and throughout, B(2n+1, 2n+1+q) denotes
an S2n+1-bundle over S2n+1+qwith attaching map ff1 a generator of ss2n+q(S2n+1)*
*, and
q = 2p - 2. Note also that the result of [5] implies that if n p, then expp(S*
*U(n)) =
expp(S3 x . .x.S2n-1) = n - 1.
Proposition 1.4. If p is odd, then expp(B(3, 2p + 1)) = p + 1, while if n > 1, *
*then
n + p - 1 expp(B(2n + 1, 2n + 1 + q)) n + p.
Proof.This just combines [3, 1.3] for the lower bound and [11, 2.1] for the upp*
*er
bound. ||
Upper and lower bounds for the p-exponents of Sp(n) and Spin(n) can be extrac*
*ted
from Theorems 1.1 and 1.2 using long-known relationships of their p-localizatio*
*ns
to that of appropriate SU(m). Indeed, Harris ([9]) showed that there are p-loc*
*al
equivalences
SU(2n) ' Sp(n) x (SU(2n)=Sp(n)) (1.5)
Spin(2n + 1) ' Sp(n) (1.6)
Spin(2n + 2) ' Spin(2n + 1) x S2n+1. (1.7)
Combining this with Theorems 1.1 and 1.2 leads to the following corollary.
Corollary 1.8. Let p be odd.
ODD-PRIMARY EXPONENTS OF LIE GROUPS 3
(1)expp(Spin(2n + 2)) = expp(Spin(2n + 1)) = expp(Sp(n))
expp(SU(2n)), which is bounded according to Theorem 1.1.
(2)expp(Sp(n)) 2n - 1 + p(b2n_pc!).
(3)If 1 t < p, and tp - t + 2 2n tp + 1, then expp(Sp(n))
2n.
Proof.The second and third parts of (1) are immediate from (1.6) and (1.5), whi*
*le
the first equality of (1) follows from (1.7) and the fact that expp(Spin(2n + 1*
*))
expp(S2n+1), which is a consequence of part (2) and (1.6). For parts (2) and (3*
*), we
need to know that the homotopy classes yielding the lower bounds for expp(SU(2n*
*))
given in Theorem 1.2 come from its Sp(n) factor in (1.5). To see this, we first*
* note
that in [2, 1.2] it was proved that, if p is odd and k is odd, then
v-11ss2k(Sp(n); p) v-11ss2k(SU(2n); p).(1.9)
These denote the p-primary v1-periodic homotopy groups, which appear as summands
of actual homotopy groups. The proofs of [7, 1.1] and [8, 1.8], which yielded T*
*heorem
1.2, were obtained by computing certain groups v-11ss2k(SU(n); p) with k n-1 *
*mod
2. When applied to SU(2n), these groups are in v-11ss2k(SU(2n); p) with k odd, *
*and
so by (1.9) they appear in the Sp(n) factor. ||
For all (X, p) with X an exceptional Lie group and p an odd prime, except (E7*
*, 3)
and (E8, 3), we can make an excellent comparison of bounds for expp(X) using re*
*sults
in the literature. We use splittings of the torsion-free cases tabulated in [3,*
* 1.1], but
known much earlier.([10]) In Table 1, we list the range of possible values of e*
*xpp(X)
when the precise value is not known. We also list the factor in the product dec*
*om-
position which accounts for the exponent. Finally, in cases in which the expon*
*ent
bounds do not follow from results already discussed, we provide references. He*
*re
B(n1, . .,.nr) denotes a space built from fibrations involving p-local spheres *
*of the
indicated dimensions and equivalent to a factor in a p-localizaton of a special*
* unitary
group or quotient of same. Also, B2(3, 11) denotes a sphere-bundle with attach*
*ing
map ff2, and W denotes a space constructed by Wilkerson and shown in [12, 1.1] *
*to
fit into a fibration K5 ! B(27, 35) ! W . Finally, K3 and K5 denotes Harper's
space as described in [1] and [11].
4 DONALD M. DAVIS AND STEPHEN D. THERIAULT
Theorem 1.10. The homotopy p-exponents of exceptional Lie groups are as in Table
1.
Table 1. Homotopy exponents of exceptional Lie groups
___X______p____||expp(X)_________Factor___________Reference_____
G2 3 || 6 B2(3, 11) [3, 1.3],[11, 2.2]
G2 5 || 6 B(3, 11)
__G2_____>_5___||___5_____________S11___________________________
F4, E6 3 || 12 K3 [1, 1.6], [11, 1.2]
F4, E6 5, 7 ||11, 12 B(23 - q, 23)
F4, E6 11 || 12 B(3, 23)
_F4,_E6__>_11__||___11____________S23___________________________
E7 5 ||18, 19, 20 B(3, 11, 19, 27, 35)factor of SU(18)
E7 7 ||17, 18, 19 B(11, 23, 35) factor of SU(18)
E7 11, 13 ||17, 18 B(35 - q, 35)
E7 17 || 18 B(3, 35)
___E7____>_17__||___17____________S35___________________________
E8 5 || 30, 31 W [6, 1.1],[12, 1.2]
E8 7 ||29, 30, 31, 32B(23, 35, 47,[59)3, 1.4],Proposition 2.3
E8 11 - 23 ||29, 30 B(59 - q, 59)
E8 29 || 30 B(3, 59)
E8 > 29 || 29 S59
2. Proof of Theorem 1.1
In [11, Lemma 2.2], the second author proved the following result.
Lemma 2.1. ([11, 2.2,2.3]) Suppose there is a homotopy fibration
q 2n+1
F ! E -!S
q* 2n+1
where E is simply-connected or an H-space and | coker(ss2n+1(E) -! ss2n+1(S *
*))|
pr. Then expp(E) r + max(expp(F ), n).
In [11, 2.2], it was required that E be an H-space, but [11, 2.3] noted that *
*if E is
not an H-space, the desired conclusion can be obtained by applying the loop-spa*
*ce
ODD-PRIMARY EXPONENTS OF LIE GROUPS 5
functor to the fibration. We require E to be simply-connected so that we do not*
* loop
away a large fundamental group. We now use this lemma to prove Theorem 1.1.
Proof of Theorem 1.1.The proof is by induction on n. Let the odd prime p be im-
plicit, and let SU0(n) denote the factor in the p-local product decomposition (*
*[10]) of
SU(n) which is built from spheres of dimension congruent to 2n - 1 mod q. By the
induction hypothesis, the exponents of the other factors are the asserted amo*
*unt.
We will apply Lemma 2.1 to the fibration
q 2n-1
SU0(n - p + 1) ! SU0(n) -!S .
q* 2n-1
In order to determine | coker(ss2n-1(SU0(n)) -! ss2n-1(S ))|, we use the cla*
*ssical
result of Borel and Hirzebruch ([4, 26.7]) that
ss2n-2(SU(n - 1)) Z=(n - 1)!.
When localized at p, it is clear that its p-component Z=p p((n-1)!)must come fr*
*om the
SU0(n-p+1)-factor in the product decomposition of SU(n-1), since ss2n-2(SU(n-1))
is built from the classes ffi2 ss2n-2(S2n-1-iq)(p). Thus
ss2n-2(SU0(n - p + 1)) Z=p p((n-1)!),
and the exact sequence
q* 2n-1 0
ss2n-1(SU0(n)) -! ss2n-1(S ) ! ss2n-2(SU (n - p + 1))
implies
p(| coker(q*)|) p((n - 1)!). (2.2)
(a.) By the induction hypothesis, expp(SU0(n - p + 1)) n - p + p((n - p)!)*
*. By
hypothesis, n-p < p2and hence p((n-p)!) p-1. Thus expp(SU0(n-p+1)) n-1,
and so by 2.1 and (2.2)
expp(SU0(n)) p(| coker(q*)|) + n - 1 p((n - 1)!) + n - 1,
as claimed.
(b.) By (a), part (b) is true if p2 + 1 n p2 + p - 1. Let n p2 + p, a*
*nd
assume the theorem is true for SU0(n-p+1). Then by Lemma 2.1 and the induction
hypothesis
_ n-p-1 !
b_____c - p + 2
expp(SU0(n)) ((n - 1)!) + n - p + 1 + p - 3 + p-1 .
2
6 DONALD M. DAVIS AND STEPHEN D. THERIAULT
Note that even if expp(SU0(n - p + 1)) happened to be less than n - 1, our upper
bound for it is n-1, and so this bound for expp(SU0(n)) is still a correct de*
*duction
from 2.1.
Since p((n - 1)!) bn-2_p-1c, we obtain
$ % _ !
n - 2 bn-2_c - p + 1
expp(SU0(n) _____ + n - 2 + p-1
p - 1 2
$ % _ ! _ !
n - 2 bn-2_c - p + 2 bn-2_c - p + 1
= _____ + n - 2 + p-1 - p-1
p - 1 2 1
_ !
bn-2_c - p + 2
= n + p - 3 + p-1 ,
2
as desired. ||
The result in part (b) could be improved somewhat by a more delicate numerical
argument.
Part (b) of the following result was used in Table 1.
Proposition 2.3. Let p = 7.
a. exp7(B(23, 35, 47)) 25.
b. exp7(B(23, 35, 47, 59)) 32.
Proof.The thing that makes this require special attention is that these spaces *
*are
not a factor of an SU(n), because they do not contain an S11. There are fibrati*
*ons
B(23, 35) ! B(23, 35, 47) ! S47
and
B(23, 35, 47) ! B(23, 35, 47, 59) ! S59.
Since, localized at 7, ss46(S23) ss46(S35) Z=7, we have |ss46(B(23, 35))| *
* 72, and
similarly |ss58(B(23, 35, 47))| 73. (In fact, it is easily seen that these a*
*re cyclic
groups of the indicated order.) Using 2.1 and that exp7(B(23, 35)) 18 by 1.4,*
* we
obtain
exp7(B(23, 35, 47)) 2 + max(18, 23) = 25,
and then
exp7(B(23, 35, 47, 59)) 3 + max(25, 29) = 32.
ODD-PRIMARY EXPONENTS OF LIE GROUPS 7
||
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Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA
E-mail address: dmd1@lehigh.edu
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24
3UE, United Kingdom
E-mail address: s.theriault@maths.abdn.ac.uk